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THE UNIVERSITY 
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8057-S 





THE INTERPRETATION 


OF 


MATHEMATICAL FORMULA 


BY 
EDWIN J. HOUSTON, Pu. D. (Princeton), 
AND 


ARTHUR E. KENNELLY, Sc. D. 


NEW YORK 
AMERICAN TECHNICAL BOOK CO. 
45 Vesty STREET 


1898 





EDWIN J. HOUSTON anp ARTHUR E. 
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PREFACE. 





Ir is wonderful how much is capable 
of being expressed by a mathematical 
formula, and how little mathematical 
proficiency is necessary for its inter. 
pretation. Nevertheless, it 1s commonly 
believed that a long course of mathe- 
matical training is essential to an inter. 
pretation of such formule as are found in 
ordinary technological text-books, The 
authors have endeavored, in this little 
book, to show that this is fallacious, 
that, on the contrary, a mere knowledge 
of arithmetic, as a preparatory training 
to a perusal of this book, will give to 
the student all the insight that is needed 


ili 


344820 


lv PREFACE. 


to understand applied mathematical 
formule. The authors of course do not 
claim, however, that those who have read 
this book thereby become expert mathe- 


maticians. 


PHILADELPHIA, January, 1898. 


CONTENTS. 


CHAPTER PAGE 
I. ADDITION, ; : ‘ : : 1 
II. Susrraction, : ‘ , 14 
II. Muvtrirrication, . ‘ ; BAe gat 
LVowIVvISIion, —. A A : : 34 
V. Invontutrion. Powers, . : Stes 
Wie voturion. Roors, . ‘ ; 61 
VII. Equations, ; ‘ : é ans (6 
VIII. Loagarirunms, ; ; : i 78 
IX. TRIGONOMETRY, : : : mes Yi 
X. Hypersonic TRIGONOMETRICAL 
Functions, . ; i A 128 
XI. DirrereEntTIAL CaLcutus, 5 kon 
XII. Inrrerart Carcutys, ‘ : 165 
XIII. DeErtTERMINANTs, d : ; . 193 


XIV. Synopsis or Sympots CoMMONLY 
Founp IN MATHEMATICAL For- 


MULA, ‘ r ; : - 2038 





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THE INTERPRETATION OF MATHE- 
MATICAL FORMULA, 





ALGHBRA. 


CHAPTER I. 


ADDITION. 


Algebra is that branch of mathematics 
which treats of the properties of numbers 
and their general relations by means of 
symbols. It may, therefore, be regarded 
as a system of generalized arithmetic. 
The principal operations dealt with in 
algebra are addition, subtraction, “multi- 
plication, division, involution, evolution, 


2, THE INTERPRETATION OF 


and the solution of the equations involv- 
ing these operations. We shall proceed 
to consider these operations in order. 

In arithmetic, if we add two numbers 
together, we .obtain their sum; as, for 
example, when we say that 5, added to 
7, gives a total of 12; or, as we may 
express it, the sum of 5 and 7 is equal 
to 12; which, again, may be expressed 
symbolically,5 +7 = 12. Here the sign 
(+) is called the plus sign, and indicates 
the operation of addition, while the sign 
(=) 1s called the equality sign, and in- 
dicates the condition of equality between 
the two things which it connects. 

The expression 


5+7=19 (1) 


is called an equation, in which the left- 
hand member consists of two terms; 
namely, 5 and 7, while the right-hand 


MATHEMATICAL FORMULA. 3 


member consists of a single term; namely, 
12. The equation is read thus: 


Five plus seven is equal to twelve. 


We may extend the terms of an equation 
to any desirable extent. Thus we may 
write the following equations which are 
obviously true, 


$8+3+6= 12, (2) 
or o+t38+6=9- 3, (3) 
or 4+34+2+1+4=103 (4) 


The number of terms on either side of 
the equation may be any whatever. All 
that is necessary is that the sum of the 
terms in the left-hand member should be 
equal to the sum of the terms in the 
right-hand member. 

The four preceding equations deal with 
terms. which are all simple numbers or 
definite numerical quantities. We may, 


4 THE INTERPRETATION OF 


however, extend the same reasoning sym- 
bolically. Thus, in the equation 


atb=ce (5) 


we have a statement that the sum of two 
quantities on the left-hand side, which are 
represented respectively by the letters a 
and 6, is equal to the quantity on the 
right-hand side, represented by the letter 
c. If we make a, equal to 5, and 6, equal 
to 7, we reproduce equation (1), and we 
are compelled to make ¢, equal to 12. In 
other words, giving definite numerical 
values to the terms on the left-hand side 
of the equation determines the value of 
the symbol on the right-hand side. There 
is this difference, however, that in an 
arithmetical equation such as appears 1n 
equation (1), 


5+7 = 19, 


MATHEMATICAL FORMULA. 5 


we are dealing with numbers only, while 
in the case of a corresponding algebraic 
equation, 

atb=<e, 


we are not restricted to numerical quan- 
tities, because although a, b, and c, may 
stand for the numbers 5, 7, and 12, respec- 
tively, and thus reproduce the arithmetical 
equation, they may also stand for other 
magnitudes which are not merely numeri- 
cal. For example, if a man, weighing a 
pounds, is suspended from a rope, and the 
man holds in his hand a weight of 6 
pounds, then the total weight which the 
rope has to sustain is @ + 6 pounds, and 
we may express this in the form of the 
equation 


c=a+b pounds, (6) 
which is the same as the equation 


a+tb=e pounds. (7) 


6 THE INTERPRETATION OF 


Thus, if the man weighs 150 pounds, then 
a = 150, and if the weight he carries in 
his hand is 10 pounds, then 6 = 10, and 
the total weight on the rope 


G2 100 =e) 
=yt Hf 8, pounds. 


Here the quantities which are considered 
in the equation are the magnitudes of cer- 
tain weights, and these magnitudes are 
treated numerically, through the use of the 
unit of weight, or a pound, by which they 
are compared numerically. Consequently, 
while an arithmetical equation considers 
equality between numbers, an algebraic 
equation may consider equality between 
any kind of magnitude, from the brilliiancy 
of a fixed star to the area of a piece of 
land. In all cases, however, the same 
kind of magnitude only can be compared 
in an equation; so that if the left-hand 


MATHEMATICAL FORMULZ. 7 


member a + 6, of equation (7) represents 
two magnitudes expressed in pounds, the 
right-hand member c, must also represent 
'a magnitude expressed in pounds. 
Moreover, in an arithmetical equation, 
such as 
5 i nae, 


the terms are all fixed in value, and the 
equation is susceptible of but one inter- . 
pretation; namely, that the sum of 5 and 
7 is equal to 12; but in an algebraic 
equation, 

at+b=e, 


the fact that the terms are symbolical, 
enables them to represent any desired 
numerical values; the only requisite 
condition being that c¢, shall be equal to 
the sum of the values of @ and 4, or that 
the equality expressed by the equation 
shall subsist, Thus, we have seen: that 


8 THE INTERPRETATION OF 


if the weight a, of the man suspended by 
the rope was 150 pounds, and the weight 4, 
which he carried, was 10 pounds, then ¢, 
the tension in pounds weight, supported 
by the rope, 1s 160 pounds; but the 
equation would also correctly represent 
a case in which the man weighed 230 
pounds, and the weight he carried 50 
pounds; but in this case the total tension 
on the rope would be 


230 + 50 = 280 pounds. 


It is evident, therefore, that the algebraic 
equation, 


at+tb=e, 


is susceptible of an infinite variety of 
interpretations, according to the values 
which are given to the terms, and is, 
therefore, of a much more general nature 


MATHEMATICAL FORMULA. 9 


than the simple type of arithmetical 
equation, 
5+7=12, 


to which it corresponds. 

An algebraic equation involving the 
operation of addition may extend to any 
number of terms, just as we have seen an 
arithmetical equation may similarly extend. 
Thus, suppose that, at a railway station, a 
truck is loaded with a number of boxes, 
and that their individual weights are repre- 
sented by the symbols a, 4, ¢, d, e, f and g, 
respectively. Then, if we expressed the 
total weight upon the truck by the letter w, 
we obviously obtain the equation 


w=at+b+e+d+e+ f+ pounds. (8) 


This is a symbolic method of making the 
following statement: The total weight 
upon the truck of the seven boxes with 


10 THE INTERPRETATION OF 


which it is loaded, is equal to the sum of 
the weights of all these boxes. 

It is evident that equation (8) may repre- 
sent an infinite variety of arithmetical equa- 
tions, each of which would be true in its own 
particular case. We have only to substitute 
for the symbols a, 4, ¢, d, é, f and g, their 
proper numerical values, and the value of 
w, their total weight, then follows from 
the equation. If the weights of the indi- 
vidual boxes are expressed in pounds, 2, 
will be determined in pounds. If the 
weights of the individual boxes are ex- 
pressed in kilogrammes, w, will be deter- 
mined in kilogrammes, and so on for any 
unit of weight. Equation (8) is, there- 
fore, an equation connecting gravitational 
magnitudes, but it is evident that this 
equation is capable of representing any 
summation relation between any kind of 
magnitude whatever, For example, if we 


MATHEMATICAL FORMUL. dah 


form an equation for the total amount of 
rainfall m a week, and represent this total 
by the letter w, where w,is a quantity 
expressed in inches of water evenly dis- 
tributed over a level surface; then if 
a, b,c, d, 6, f and g, represent the rainfall in 
inches, occurring on each of the seven days 
of the week respectively, the equation (8) 
is no longer an equation between gravita- 
tional magnitudes, but an equation between 
rainfall magnitudes, and is susceptible 
again of infinite variety of interpretations. 
It is evident that the same algebraic equa- 
tion would be true for any week in the 
year, but the actual numerical solution of 
the equation, obtained by substitution in 
the terms on the right-hand side, would, 
probably, vary considerably from week to 
week. 

Similarly, if instead of taking the weekly 
rainfall, we desired to take the monthly 


12 THE INTERPRETATION OF 


rainfall, we might construct a similar equa- 
tion of 30 or 31 terms, according to the 
month selected, and make their sum still 
equal to w; but the quantity represented 
by w, would no longer be a weekly rain- 
fall, but a monthly rainfall. Again, we 
might make an equation of 365 terms, each 
term representing the rainfall during one 
of the days of the year, and equate the 
whole series to w. Here w, would be no 
longer a weekly rainfall, but a yearly or 
annual rainfall. A number of the terms 
on the right-hand side might be numerically 
equal to zero, representing the fact that on 
those days there was no rainfall; the other 
terms would be numerically expressed in 
inches and fractions of an inch. In such 
eases, if the letters of the alphabet are 
not sufficiently numerous to supply all the 
terms, the letters of the Greek alphabet, or 
the German alphabet, may be used, or 


MATHEMATICAL FORMULA. icy 


capital letters may be employed, or 
suffixes may be used. Thus, equation (8) 
might be written if desired 


Pera Ps te A eB boa? +O}. 


Here the first two terms on the right-hand 
side are the letters Alpha and Beta, of the 
Greek alphabet, the third term is the letter 
d, of the German alphabet, the fourth and 
fifth terms are English capitals, while the 
sixth and seventh terms are @ prime, and 0 
prime. It is not, of course, usual to adopt a 
heterogeneous symbolism of this character, 
and, as a rule, letters of the English alpha- 
bet, either small, capital, or suffixed, are 
the most generally employed. 


CHAPTER IL. 
SUBTRACTION. 


Ir we diminish the value of a certain 
number, say 12, by another similar number, 
say 5; or, as it is generally called, if we sub- 
tract 5 from 12, we obtain the difference 
7. This may be expressed in language by 
the following equation: 


Twelve less five is equal to seven. 
Or, in numbers, 
12-5 =%. (1) 


Here the sign (—) is called the minus sign, 
and indicates that the quantity which fol- 
lows it is to be subtracted from the quan- 
tity with which it is associated. 

14 


MATHEMATICAL FORMULA. 15 


This equation may be generalized alge- 
braically in the following manner : 


a—b=e, (2) 


which makes the following statement: If 
from the quantity a, we subtract the quan- 
tity 6, the remainder will be the quantity c. 

In the early stages of arithmetical 
science, subtraction could only be per- 
formed when the number to be subtracted 
was less than the number from which the 
subtraction was to be made. ‘Thus, it was 
considered impossible to subtract six from 
five. Later development, however, showed 
‘that by an extension in meaning of the 
term subtraction, such an operation could 
be readily performed, if it was considered 
that the remainder was negative, so that the 
equation 

5-6=-1 


16 THE INTERPRETATION OF 


represents that if six be taken from five, 
the remainder is minus one. Or, equation 
(2) holds good whether 4, is greater than, 
equal to, or less than a. For example, sup- 
pose that a, represents a bank account of 
five hundred dollars, and suppose that 8, 
represents a draft made on the account. 
If 4, 1s less than a, the remainder ec, will be 
a positive balance in dollars at the bank. 
For example if @ = $500, and 6 = $300, ¢, 
will be $200, but if 4, is greater than a, say 
$550, then the account at the bank will be 
overdrawn, and will leave a negative bal- 
ance. Thus, 


500 — 550 = — 50, 


so that the account will owe $50 to the bank. 

Any quantity to which the minus sign 
is prefixed may, therefore, be considered as 
a negative quantity, and from this point of 
view subtraction is only an extension of 


MATHEMATICAL FORMULA. ce 


the operation of addition, except that, 
whereas simple addition only adds positive 
terms, subtraction, included in addition, 
considers the addition of terms which are 
both positive and negative. 

For example, if we consider the equation 
of the preceding chapter, 


c=a+), 


where a, is the weight of a man who is 
suspended from a rope; 4, 1s the weight of 
an object which he holds in his hand; and 
c,1s the total weight on the rope. Here, 
b, instead of being a weight, may be a nega- 
tive weight, or a sustaining power.. For 
example, if he held in his hand a balloon 
which exerted an ascensional pull of say 
50 pounds, then 6, would be — 50, and if 
his own weight were 150 pounds, 
Gu pO 50 
100 pounds, 


I| 


18 ' THE INTERPRETATION OF 


Here 0, is a negative quantity. Again, 
if the balloon exerted an ascensional pull 
greater than the weight of the man, say 
200 pounds, then the weight which the 
rope has to sustain would be 


= 150 = 200 
= — 50 


o 


or, c, would be a negative, or ascensional 
pull, so that the man and rope would be 
pulled upward. Consequently, in any 
equation such as 


Ci Ot) Gare 


any of the terms a, 0, or c, may happen to 
have negative values, and the addition 
would then have to be made with these 
considerations in view. For the negative 
terms would have to be subtracted from 
the positive terms, and if they exceeded 


MATHEMATICAL FORMULA. 19 


the latter, the sum a, would remain nega- 
tive. Thus, suppose the equation to 
represent an account at the bank, and that 
- four payments have been made to the 
account. Then the equation is equivalent 
to the statement that the total payment is 
equal to the sum of the separate payments. 
Suppose, however, that @ and 6, are pay- 
ments which have been made to the credit 
of the account, and that ¢ and d, are nega- 
tive payments, or payments which have 
been made at the expense of the account. 


Then if 


a = $100, 56 = $200, c = $300, and d 
= $400, 


the equation becomes : 


] 


100 + 200 — 300 — 400 
300 — 700 
— $400. 


a 


20 MATHEMATICAL FORMULA. 

So that the total payment is a negative 
or debit payment of $400. With this 
view of negative quantities, it 1s evident 
that simple addition and subtraction fall 
under one set of operations. 


CHAPTER III. 
MULTIPLICATION. 


In arithmetic, when two numbers are 
multiplied together the result is called 
their product. ‘Thus, if we multiply 5 by 
7, we obtain 35, as the product. ‘This is 
expressed symbolically as follows: 


bi X 7 = 35. 
Here the sign (x) is called the multipli- 
cation sign, and is read: “multiplied by,” 
so that the equation reads: 


Five multiphed by seven is equal to 


thirty-five. 
Expressed in algebraic symbols: 
enU) 10 
or, el pnp eae 


21 


34 THE INTERPRETATION OF 


In these equations c, stands for the prod- 
uct of a and 6, whatever those symbols 
may represent. Tor example, if a, repre- 
sents in dollars and cents the wages of a 
laborer per day, and 6, is the number of 
days’ work done by the laborer at this pay, 
then ¢, is the total wages due to him at the 
end of ddays. Thus,ifa = $1.60 and 6 = 
6 days, then the amount due to the laborer 
is 
c = 1.60 X 6 
= 9.60 dollars. 


It is evident that the equation 
grit it 


is susceptible of an infinite variety of inter- 
pretations, according to the values which 
may be assigned to a and 6. It is to be 
noticed that if @ and 0, are mere numbers, 
or numerical quantities, their product is 


MATHEMATICAL FORMULA. pay 


also a number, but if @ and 8, are both 
physical magnitudes, then their product ¢, 
will not be the same kind of quantity as 
either of the components. Thus, suppose 
that a weight of a pounds, is raised through 
a vertical distance of > feet, then we know 
that the product ax 4, is equal to the 
work done in the process of lifting, and is 
expressed in units of work called foot- 
pounds; one foot-pound being the amount 
of work required to raise one pound through 
a vertical distance of one foot against ravi- 
tational force. Consequently, this equation 
would read : 


@ (pounds) x 6 (feet) = ¢ (foot-pounds). 


Here it will be seen that the kind of 
magnitude on the right-hand side of the 
equation, namely, foot-pounds or work, 
is different from either of the two magni- 


d 


24 THE INTERPRETATION OF 


tudes appearing on the left-hand side; 
namely, weight and distance. 

The multiplication sign is frequently 
omitted between two symbols which are 
to be multiphed together. Thus, the 
above equation may be written 


c= ab. 


Here the period takes the place of the 
multiplication sign. Or, we may write 


C=", 


both multiplication sign and period being 
omitted; the symbols being merely 
written after each other, and, since the 
order of the two quantities to be multi- 
plied is indifferent, we have also 


Gu Ud, 


where the multiplication sign is likewise 
omitted. The two quantities a@ and 8, 


MATHEMATICAL FORMULA. 25 


being written side by side indicate that 
their product is to be taken. Similarly, 
any number of quantities written in suc- 
_ cession indicate that their continued prod- 
uct has to be taken. For example, 


w= gac 


ig an equation which means that the 
quantity w, 1s equal to the product of 
the three quantities represented by the 
symbols g, a@ and ¢, respectively. This 
equation could be written: 


C190 
or, Gee. C, 
Thus, if 
g—2,¢ — 4, and¢ = 6 
we = 48. 


It will be evident that in any equation, 


such as 
e=atb—-~e 


26 THE INTERPRETATION OF 


in which the three terms on the right- 
hand side are represented as simple quan- 
tities, any of these terms may stand for 
products. Thus a, may be the product 
of e and f, or 


a=; 
6, may be the product of h, & and 4, or, 
b= hig; 


while ¢, may be the product of p, g, 7 and 
8, OF, 


C = pgrs. 


Consequently, such an equation may be 
written 


e= eof + hky — pars. 


If the values of ¢, f, h, 4,7, p, q, 7, & are 
all given, the value of a, can be deter- 
mined by first multiplying e and f, 
together, then multiplying h, & and 9, 


MATHEMATICAL FORMULA. 27 


together, then multiplying p, g, 7 and s, 
together and, finally, adding the three 
products so obtained, the last quantity 
being considered as negative. 

In the arithmetical equation 


Boe 60 ocr ex 1, 


in which each of the terms on the right- 
hand side is a product of a number and 
number 7, it is clear that the same result 
-will be obtained if we add the numbers 
by which seven is to be multiplied before 
making the product, or we may write 
the equation: 


63 = 7 X (5 +8 41) 
= 1X9 
= 63. 


Herethesymbol( __), called the brackets 
or parentheses, indicates that all of the 
quantities within them are to be operated 


98 THE INTERPRETATION OF 


upon by the multiplication sign. In other 
words, the three quantities within the 
brackets are to be grouped together and 
considered as a single quantity. 

From the equation 


68=7x (6+34+1) 
we also derive the equation 


6= 385+ 214+7 
=" 63, 


The same rules apply algebraically. Thus, 


from the equation: 
w=a(b+e4+d), 


which means that the quantity a, 1s equal 
to the product of the quantity a, multi- 
plied by the compound quantity which 
is the sum of 0,¢ and d, we derive the 


equation 
«= ab+ac-4 ad, 


“MATHEMATICAL FORMUL2. 29 


from which we see that when a factor a, 
appears outside of the bracket containing 
several terms, the factor may be considered 
- to apply to each of these terms in suc- 
cession. 

Thus, suppose that a passenger steamer 
has accommodation for / first-class passen- 
gers, § second-class passengers and g 
steerage passengers, and that the ship 
makes ten journeys with every berth 
filled. Then if a, be the total number of 
people transported in all, in the 10 trips, 
we have the equation 


10f + 10g + 10s 
or, Di =A) ire geste 8), 


| 


where the compound term or quantity in 
the bracket, namely, + 7 + s, is the total 
number of passengers in one trip. Simi- 


Jarly, in n, trips where m, is any number, 


30 THE INTERPRETATION OF 


the total number of passengers carried will 
be . 
ew=n(f+g+t+S8). 


Sometimes a straight line or vinculum is 
used to connect a number of terms into a 
single group. With the aid of the vincu- 
lum the last equation would be written 


xew=nxftgt+s. 


In a similar manner, several compound 
terms may be associated together into a 
product. Thus: 


w=(a+b+c)d+e4+/) 


means that the quantity a, is the product 
of two compound terms, the first term 
being the sum of a, 0 and e, while the 
second is the sum of d,e and f. Here if 


Arar sorte 
and Moyes Uber toa wd 


MATHEMATICAL FORMUL®. 31 


then, from the equation, 
es eS 


For example, if an elevator rises through 
three stages, the first of which is a, 
feet, the second 6, feet, and the third «, 
feet, and carries three passengers, the first 
of whom weighs d, pounds, the second e, 
pounds, and the third f#, pounds, then the 
total work done in lifting the three passen- 
gers through the three stages is repre- 
sented by the above equation. 

iia 30 feet, > — 20 feet, and c = 20 
feet, while d = 150 pounds, ¢ = 100. 
pounds and f = 120 pounds, we have 


8 
| 


(30 + 20 + 20) (150 + 100 + 120) 
foot-pounds 

= 70 X 370 

25,900 foot-pounds. 


32 THE INTERPRETATION OF 


As an example of multiplication in 
aleebra, we may consider the rule for 
determining the sum of an arithmetical 
series; 7. @., a Series whose successive terms 
have a constant difference, such as the series 


or 8,16. 24) 625 (40) oeeme 


This formula for determining the sum of 


an arithmetical series is 
S =n (21 — (Nie 


where S,is the sum required, V is the 
-number of terms summed, d is the com- 
mon difference between any pair of suc- 
cessive terms, /, is the last term, and 2, is 
half the number of terms. 

Thus, considering the first series 


i: 3 D 7 9 Lis 


MATHEMATICAL FORMULA. 38 


= 8 [2 X11 (621) 2) 
3 [22 — 10] 

3x 19 

at 


The sum required is therefore 36, and 
we find by actual summation that 


Peto, 10 ee eae — 86. 


CHAPTER IV. 
DIVISION. 


In arithmetic if one number is divided 
by another number, the result is called 
the quotient. This, if 15, be divided by 5, 
the quotient is 8. Or, in symbols, 


15 


sae: 


The sign of division is also employed to 
represent the operation. Thus, 
16+ 5 = 3, 
where the sign (+) is called the sign of 
division and is read: “ divided by.” 
Similarly in algebra, the equation 


C=b>.4a 
34 


MATHEMATICAL FORMULA. 35 


means that ¢, is a quantity which is equal 
to the quotient of the quantity 6, divided 
by the quantity a This equation might 
_ be written either 


b 
¢= — 
a 
or 
c = b/a. 
In one case the division bar is used to 
separate the numerator 0, from the denom1- 


nator a, of the fraction ace while in the 
| a 


second case, the line is written diagonally, 
or as a solidus. Thus, if a= 10 and 6 


= 17, 


aaa hyenas: 


The process of division may be extended 
to compound terms which are contained in 


brackets. Thus, 
e©=(a+6—c)+(d+e) 


36 THE INTERPRETATION OF 


means that the first compound term, which 
is the sum of + a, + J and — «, is divided 
by the sum of the quantities d and ¢, or 
_at+b—-e 
el i 


If a+ b —«, be represented by A, and d 
+ e, by B, then 
pe os or A= ah. 

The operations of multiplication and 
division may be readily associated in an 
equation. For example, 

Re? (wire). 
c+td 


Here the quantity a, 1s stated to be equal 
to a quantity e, multiplied by a fraction, 
the numerator of which is the sum of 
three products; namely, the sum of ay, cd 
and ef, while the denominator is the sum 


MATHEMATICAL FORMULA. of 


of the quantities cand d. If we suppose 
that the values of a, ¢, d, e, f and y, are all 
given, we may proceed to simplify the 
- equation by forming the numerator and 
expressing it by the quantity A, and then 
forming the denominator and expressing 1t 
by the quantity 6. The equation will 
then be 


A 
Gb gman ty kee 


Ls 
eA 
=: 


It will be found that a great number of 
formule or algebraic rules for the determi- 
nation of unknown quantities only involve 
operations of addition, subtraction, multi- 
plication and division. For example, in 
calculating the expansion of a gas, refer- 
ence is usually made to the following 
formula: 


V,= V, 1 + at), 


88 THE INTERPRETATION OF 


where Vp, is the volume occupied by a gas 
at the temperature of melting ice, or zero 
degrees Centigrade, V,, 1s the volume 
occupied at some other temperature ¢ de- 
orees Centigrade, and a, is a coefficient or 
constant numerical multiplier which is 
0.00366. The equation is, therefore, equiv- 
alent to the following statement: “ ‘The 
volume of a gas at a temperature 7° C. is 
equal to its volume at zero Centigrade, 
multiphed by a quantity which is the sum 
of unity and the product of the temper- 
ature ¢ C. and 0.00366. Thus, if the 
volume V, = 300 cubic feet and ¢ = 15° 
C., then the volume of the gas at 15° C. is 


V, = 300 (1 + 0.00366 X15) 

300 (1 + 0.0549) 

300 X 1.0549 

316.47 or 316 1/2 cubic feet, 


approximately. 


MATHEMATICAL FORMULA. 39 


Again, in the discussion of electric cir- 
cuits, in which two resistances of 7, ohms, 
and 7, ohms, are connected in parallel, then 
-it may be shown that the joint resistance 
Le, of the pair is 


"1 Ve 
R= - — ohms. 
Tiss he 





Thus, if 7, = 5.5 and 7, = 3.7, then ZR, the 
joint resistance, 1s 

D,0 ee our 

O-O i eet 

20.35 


2 


te 





v. 
= 2.212 ohms approximately, 
or 
212 


1000 





ohms. 


In the theory of the conduction of heat, 
the following equation occurs : 


Q=K'=t ar 


40 THE INTERPRETATION OF 


Here, Q, stands for the quantity of heat 
which passes in a given time, Z’seconds, 
through a slab of material whose thermal 
conductivity is A, and having a surface 
area of A square centimetres, and a thick- 
ness of ¢d centimetres, one face of the slab 
being maintained at a temperature ¢° C. 
and the other at a lower temperature ¢° C. 
Then the equation asserts that the flow of 
heat 1s equal to the product of the conduc- 
tivity “XY, the difference of temperature 
t’ — t, between the faces, the surface area 
A, and the time 7} divided by the thick- 
ness d. If we suppose a slab of copper 
whose conductivity A = 1.08, the surface 
area of the slab being A = 750 square 
centimetres, its thickness d = 2 centi- 
metres, and the difference of temperature 
between the faces of the slab (¢ — ¢) = 
5° C. then the flow of heat in the time 
Z’= 10 seconds will be 


MATHEMATICAL FORMULA. 41 


mpl:03:x.5 x 9750 X 10 
= 


fe 


thermal units. 


@ 


= 19,312.5 


It is important to observe that much 
may be learned from the form of an equa- 
tion concerning the nature of the quanti- 
ties with which it deals, without actually 
computing or solving it. .Thus, in the 
preceding equation, we observe that the 
flow of heat passing through a slab of 
uniform material, increases directly with 
the time 7} because the symbol 7} appears 
as a factor, so that if we double the value 
of Z} we necessarily double the value of Q. 
Again, the value of @Q, increases directly 
with the active area through which the 
conduction of heat takes place, because 4, 
appears also as a factor in the product. 
Similarly, the value of @, increases directly 
with the difference of temperature ¢'—4¢, 
between the faces of the slab. On the 


49 THE INTERPRETATION OF 


contrary, however, it will be noticed that 
the thicker the slab, or the greater the 
value of d, the smaller will be the value 
of Y; or, in other words, that Q, varies 
inversely with the thickness d. 

In the mensuration of solids we find a for- 
mula for obtaining the surface of a right 
cylinder in terms of its radius and height 


S = Inr(h + 1) 


where Sj is the surface of the cylinder in 
square inches, including base and top; 7, 1s 
the radius of the cylinder in inches; A, is 
the height of the cylinder in inches; and z, 
is the numerical ratio of the circumference 
of a circle to its diameter, or, approxti- 
mately, 3.1416. 


Consequently, if 7 = 2 and 2 = 10, 

§ = 2 X 38.1416 x 2 (10 + 2) 

2 X 8.1416 x 2 X 12 

= 150.797 square inches, approximately. 


MATHEMATICAL FORMULA. 43 


The formula for determining the horse- 
power of a single-cylinder engine from an 
indicator diagram is as follows: 


A 
P= ane horse-power. 


Here, the horse-power /, exerted upon one 
end of the piston, is equal to a fraction 
whose numerator is the continued product 
of the active area <A, of the piston in 
square inches, the mean effective pressure 7, 
during one cycle or revolution, in pounds 
per square inch, the number of revolutions 
per minute /?, and Z, the length of the 
stroke in feet and decimals of a foot. 
Thus, if a diagram taken from a cylinder 
shows that the mean pressure at the back 
of the cylinder is say 25 pounds per square 
inch, the length of the stroke Z = 3 feet, 
the number of revolutions per minute 
Ft = 160, and the active area of the back 


44 THE INTERPRETATION OF 


of the piston was 120 square inches, then 
the horse-power of the back of the cylinder 
Is: 


120 X 25 X 140 X38 


P= 33,000 
_ 1,440,000 
~ ~ 33,000 
= 43.63 horse-power, 
approximately. 


It is shown in treatises on the theory of 
probabilities that if the probabilities of the 
truth of a statement made by witnesses 
are respectively 7, Pe, 3, etc., the probabil- 
ity of the truth of a statement made by all 
the witnesses concurrently is 


fee Pi Pe «+++ Pn 
Pr Pe Pa +(1—p,) (1—py) (1 ps) ee 


This equation states that the probability 
that the event actually occurred, 1s a 


MATHEMATICAL FORMULA. 45 


fraction, of which the numerator is the 
product of the probabilities of true declara- 
tion of the respective witnesses, and whose 
denominator is the sum of two compound 
terms. The first is the product of the 
probabilities of true declaration as in the 
numerator, and the second term is the 
product of each probability subtracted 
from unity into all the others similarly 
subtracted. ‘Thus, if the probability that 


A’s statements are reliable is 3 in 4 or 2 = p, : 
that B’s as 33 TET SAID ety 1 
“ Og “ & 5“6% 5 =p 
“ D’s 66 “ LS Ry date & at yd 


then, if A, b, Cand JL, all independently 
concur in stating that a certain event 
happened, the probability that it actually 
happened, so far as it can be gauged from 
this circumstance, is 


46 THE INTERPRETATION OF 


PX 4X #x4 

EXExX 4+ (1-9 0-9 0) 0-® 
0 
0 








The probability is, therefore, 360 in 361, 
or 3860 to 1 that the event actually 
occurred. 

If cannon balls are piled in the form of 
a square pyramid, the number in one edge 
of the base being x, the total number of 
balls 4, in the completed pile is 


N = % (n+ 1) (2n + 1). 
hus; if 7,18 
NW = ¥¥ (19) (37) 
tr Ge Oe 
= 4218, 


If along rod whose weight is very small 
be loaded with weights m,, mg, ms, etc., at 


MATHEMATICAL FORMULA. 47 


distances measured from one end of the rod, 
equal to a, @, #3, etc., then it 1s shown in 
treatises on statics that the centre of 
gravity of the loaded rod is situated ata 
distance #, from the end of the rod, 
expressed by the formula 


_ My % + Me A% + Ms %y + 
m + m,+ m+ .... 





or #, 1s a fraction whose numerator is the 
sum of the products formed by each weight 
into its distance; while the denominator 
is the sum of the weights or the total weight 
of the loaded rod. 

This is sometimes written 


MA 
eae 





where Sm means the sum of all terms of 
the type m, #, and Sm means the sum of 
all terms of the type m. 


48 THE INTERPRETATION OF 





UTA Cb wes 107, , = 5 inches 
ein Co = hee 
Ms = 4 OZ. i, =A 
Mg = IEO0Z: C,; = ie 
Then 
ae 3X5 + 2X10 4+ 4x12 eee 
Bie mad jt em 
210 +h 20.443 ee 
8 Die ee 
99 


0 a 9.9 inches from the end. 


oT 


It may be shown that the probability p, 
of dealing all the 13 cards of one suit 
to one player from a whist pack of 52 
shuffled cards among four players is 


1381 x 398 
52! 


Here p, the probability is equal to a frac- 
tion, the denominator of which is the- 
factorial of 52; 2. ¢, the product 52, 51. 
50, 49, ete., 5, 4, 8, 2,1; or the continued - 


MATHEMATICAL FORMULA. 49 


product of all the numbers between that 
number and unity inclusive, while the 
numerator is the product of the factorial 
. of 13 and the factorial of 39. A factorial 
is represented by the note of admiration 
following the number. In this case 


aa 
P = 635,013,559,600 ’ 


or one chance in about 635 billions. 


CHAPTER V. 
INVOLUTION. POWERS. 


Ir we multiply any number, say 5, by 
itself, we obtain what is called its sguarz 


or 
5 xX 5 = 25, 


so that 25, is the square, or the second 
power of 5. 

Similarly, if we multiply a number by 
itself twice in succession, we obtain what is 
called its cwbe, or 


5X5 X 5 = 195, 


so that 125, is the cube, or the third power 
of 5. 


MATHEMATICAL FORMULZ. 51 
In the same way 
exe XD iees Oe = OUD 


is the fourth power of 5, and 5X 5...” 
times in succession is the nth power of 5. 
A. special notation is employed to repre- 
sent powers briefly and conveniently. 


Thus 
5? = 25, 
where the small number appended is 
called the index, or the exponent, or the 
power of 5, and the equation is read, 5 
squared, or raised to the second power, is 
equal to 25. 
In the same way 


7 = 49, because 7 X 7 = 49. 
Similarly, 
5% = 125, because 5 X 5 X 5 = 125, 


and the equation is read, 5, raised to the 
third power, or cubed, is equal to 125. 


52 THE INTERPRETATION OF 


Similarly, 16? = 4,096, or 16, raised to 
the third power is equal to 4,096. 
Sunilarly, 


i) 
125, 
and so on for all powers of 5, 


or 5" = nth power of 5. 


This equation may be generalized alge- 
braically as follows: 

x = a° = nth power of a. 
Thus, if 


a@ = 2 and’ = 3 tome 


If 
= 256 and n= 2; w = 256*/==30ogem 


If we multiply two powers together, as, 
for example, when we multiply 


5? X 5% or 25 X 125, 
the product is 5° or 3,125. 


MATHEMATICAL FORMULA. 53 


Here the exponent, or index of the prod- 
uct, is obtained by adding the exponents or 
indices 2 and 3, 


or 5? X 53 = FC +9 = 5°, 


This rule is of general application. The 
index or exponent of a product of two 
powers is always equal to the sum of their 
individual indices or exponents. This 
rule is expressed algebraically as follows: 


Mb x ge x ae ros gee + b +o) 


where «a, is any number; a, 0, and ¢, are 
the powers of a, while their product is a, 
raised to the sum of those powers. For 
example, | 


10! x 102 x 10° = 10¢+2+9 = 10° = 
1,000,000, 


or, 10 X 100 X 1,000 = 1,000,000. 


54 THE INTERPRETATION OF 
It is evident that 
10' = 10, 10* = 1,000, 
107 = 100, 10* = 10,000, 


and generally 
10° = 1 followed by x. zeros. 


From the law of addition of indices in 
the formation of products, 1t follows that 


10? X 10° = 10¢T® = 10° 


so that if we multiply 10’, or 100 by 10°, 
we obtain the same quantity, or 100, but 
this is what we would obtain if we mul- 
tiply 10° by unity. Consequently, we 
infer that 10° = 1. This is a general rule, 
expressed algebraically as follows : 


w° = 1, whatever number 2, may be. 


By the foregoing rule of the summation 


MATHEMATICAL FORMULE. 5D 


of indices in the formation of products, it 
follows that 


eS hee bee Ii 


Consequently, if we multiply 5? by 5-? we 
obtain unity as the product. But if we 


multiply 5° by ale v7. @., the reciprocal of 
x 


53, we obtain unity as the product. Con- 
sequently, 


This rule is of general application, and 


1 i: 
10-* = reciprocal of IO*.= 10° = 100’ 


and, generally, 


whatever numbers @ and nm, may be. As 


56 THE INTERPRETATION OF 


an example of the use of positive and nega- 
tive indices, we may take the following: 

The frequency of oscillation correspond: 
ing to the limits of the visible spectrum; 
z.é., the number of vibrations per second 
in the ether producing visible light, he 
between, approximately, 894,000,000,000, 
000 and 762,000,000,000,000 double vibra- 
tions per second. .These may be ex: 
pressed as 


8.94 x 10“ and 7.62 xX 10%; 


a. é@., between 3.94. (10 xX 10° (eee 
times in all, or 1 followed by 14 zeros, 
and 7.62 (10 x 10 . . .) 14 times in all, 
or 1 followed by 14 zeros. 

Again, the wave-lengths in free ether cor- 
responding to these frequencies are con- 
tained between the limits 


0.000076 and 0.00004 centimetres, 


MATHEMATICAL FORMULZ. 57 
which may be expressed 7.6 xX 10-> and 
4 X 10~° centimetres ; 7. ¢,, 


1 1 7.6 


7.6 x 10° — 7.6 


* 700,000 ~ 100,000 
centimetres, and 


1 1 + 


4X 793 = * X 700,000 = 100,000 


centimetres. 

The following table will still further 
illustrate the subject of positive and nega- 
tive indices: 


10°=1; 10'= 10; 10? = 100; 10°?= 1,000, 
etc. 
10-? = 0.1; 10-* = 0.01; 10-* = 0.008, 


etc. 
It is important to observe that the 
property of the summation of indices in the 


58 THE INTERPRETATION OF 


formation of the product only applies to 
powers of the same base. Thus, 


T= ie 


because the indices 5, 8 and 2, in this 
equation are the indices of a common base ; 


z.é, 7; but it would obviously not be true 
of 


PX Ds = ore 


because here the powers are of different 
bases ; namely, 5 and 7 

If we allow a stone to fall from the 
hand toward the earth, the formula which 
expresses the distance, through which it 
will fall in a given time ¢ seconds, is 


feet. 
This equation is equivalent to the state- 
ment: The vertical distance in feet 


MATHEMATICAL FORMULA. 59 


through which the stone will descend after 
¢ seconds have elapsed from the moment of 
release of the stone, will be the product of 
‘the quantity g, into the square of the 
time ¢ seconds, divided by 2; g, 1s known 
to be, approximately, 32.2 feet per second 
per second, so that the equation becomes 
32.90 

noe 

= 16.17 feet. 


— 
—= 


If ¢=1,2 = 1x 1=1 ands = 16.1 feet. 


Ifitz=2,f=2 x2=4 ands = 644 
feet. 


lee — 1.5, = 1.5 X 1.5 = 2.25 and 
gs = 86.225 feet. 


Again, the volume of a sphere of radius 


7 feet, is known to be 


3 
V= 28 cubic feet ; 







60 MATHEMATICAL FORMULA. 


where V,is the volume in cubic feet and 
z, is the ratio of the circumference to t 
diameter, or 3.1416, approximates 
Consequently, 
4X 8.1416 7 
3 4 
4.1888 7°. 


jis 


vier Lhe ater V = 4.1888 (1 Xa 1) 
= 4.1888 cubic feet. 


If 7 = 2, V = 4.1888 (2 X 2 « 2) = 
33.5104 cubic feet. 


Pa > 


a2 
If r = 1.75, V = 4.1888 (1.75 x 1.75 x 
1.75) = 4.1888 Xx 5.3594 = 22.45 cubic 
feet. 








CHAPTER VI. 
EVOLUTION. ROOTS. 


WE have seen that ¢nvolution consists 
- inraising a quantity, say a, to some power, 
or performing the operation 


1 6 mee EF Pe 


- Hvolution consists in reversing this, or is 
the inverse of the above operation. Thus, 
if we know that 5? = 25, or that 25, is the 
square of 5, we determine by the process 
of evolution that 5, is the number whose 
square is 25; 5, is then said to be the 
square root of 25. In the same way, 
having given the relation by involution, 


5? = 195, 


61 


62 THE INTERPRETATION OF 


evolution shows that the number 5, is the 
cube root or the third root of 125. In the 
same way 2, is the square root of 4, be- 
cause 27 = 4; 2 is the cube root of 8, 
because 2? = 8; 3 1s the fourth root of 81, 
because 3* = 81, and so on. 

A root of a number is represented sym- 


bolically by a radical sign ¥. Thus 
a= VA n 


means that a, is the cube root of the quan- 
tity n, so that 
ee, 
Similarly, 
Aiain 


means that a, is the mth root of the quan- 
tity n. The equation 


Oh) Te, 


MATHEMATICAL FORMUL®. 63 


or a, is equal to the square root of n, is 
often written 


Gi Vales 


that is, the superscript 2, 1s omitted in 
the radical sign. Thus, the equation 


a= / 64 


means that a, is the number whose square 
is 64, and, consequently @ = 8. When 
the expression whose root is to be ex- 
tracted 1s a compound term, a line or 
vinculum is placed over it, or brackets 
areemployed. Thus, 


@= V 32 + 32, ora = V(82 + 32) 


are equivalent to a = 8. 

As an example in evolution, or the ex- 
traction of roots, the following case may 
be considered. The formula which gives 
the period of time of complete vibration 


64 THE INTERPRETATION OF 


of a simple pendulum, making extremely 
small oscillations, is 


Vee 
iY q seconds. 


This equation is equivalent to the follow- 
ing statement: The time Z} occupied by 
a pendulum in making one complete to- 
and-fro motion, of indefinitely small amph- 
tude, is the product of 27, or 6.2832, and 
the square root of a ‘fraction whose numer- 
ator is the length of the pendulum and 
whose denominator is the intensity of 
gravity at the location considered. Thus, 
if the length of the pendulum be 8.05 
feet, and g, the intensity or acceleration 
of gravity, be 32.2 feet per second! per 
second, then 


8.05 
32.2 


MATHEMATICAL FORMULA. 65 


= 9 xX 3.1416 tie 
4 


ox Bt 6 ee 
= 3.1416 seconds. 


It is evident that 
4=VE 


because 


@P=axgah 


Again, from a known relation, whose dis- 
covery 18 ascribed to Pythagoras, between 
the lengths of the sides of a right-angled 
triangle; namely, 

Ilypothenuse = Y (Perpendicular)? + (Base)? , 
if we have a triangle OAD, Fig. 1, which 
contains a right angle or 90° at A, then 

OB =v(UA)* + (AB). 
If OA = 4 feet and AL = 3 feet 


C60 THE INTERPRETATION OF 


OB = v4 + 3? 
=Vv1l6+9 
= V25 
= 5 
because 5, is evidently the square root of 
25, since 5? = 25, 





Fig. 1. 


Another convenient method of express- 
ing roots or radicals consists in the em- 
ployment of fractional indices. ‘Tlrus, 
from the general law of the addition of 
indices or exponents in the formation of 
powers, we have: 


10? x 10! = 10@+ = 10! = 10, 


MATHEMATICAL FORMULE. 67 
so that 10!, multiplied by itself, or squared, 
gives 10; or, 

(10?)* = 10, or 103, is the square root of 10. 
Similarly, 

10! X 10! x 10! = 10¢+!+) = 10! = 10, 
so that 10! = V10. 


This is capable of generalization, algebra- 
ically, in the formula 


1 n ,—— 
ax = Va = nth root of the number 2. 


As an example, 9? = 3 because 3 X 3 = 9. 

We have hitherto considered powers 
which were formed with indices which are 
whole numbers or integers, but it 1s now 
easy to see what a fractional power means. 
For example, by the law of the addition of 
indices, 


TOS ic OEE 2 = 71 08 


68 THE INTERPRETATION OF 
so that 
10? = V 108 

or, the square root of the cube of ten; 

10? = 10! x 10! x 10! 

10! = (V10)’; 
or, the cube of the square root of ten. 
Consequently, 

10''= (V10) = vier 

or, the cube of the square root of ten is 
equal to the square root of the cube of ten, 
aud generally, 


m 
m 


A (S} n _— 
Asay ta ean 
(4) n/\m 
= ge = (Va)”. 
If then we take a power a*, and divide its 


index by some quantity, say 3, we obtain 
a, which is the cube root of a*, and again 


MATHEMATICAL FORMULA. 69 


if we multiply its index by any number, 
say 4, we obtain a, which is the fourth 
power of a = (a*)*. | 

Again, if we have an equation 


m= git 


the equation means that a, is a quantity 
which is equal to the 15th root of the 
11th power of a, or to the 11th power of 
the 15th root of a; 7. e., 


C= (ate)"! — V (ay 
= Ya" = @y 


CHAPTER VII. 
EQUATIONS. 


Aw equation is an algebraic expression 
of equality between two quantities, em- 
ploying the sign =. Equations differ 
almost infinitely in nature, complexity, and 
length. It is impossible, within the limits 
of this little work, to devote sufficient 
space to the subject of the treatment or 
manipulations of equations, so that the 
solution of any given equation may be 
obtained. It will be sufficient if the 
student obtains from this book a clear 
understanding of the meaning of the state- 
ment contained in any equation, so as to 
be able to interpret its signification. There 


are, however, a few general and simple 
70 


MATHEMATICAL FORMULA. a | 


rules which may be set down as a guide 
for the student in dealing with equations. 

Equations may be divided into the fol- 
lowing classes : 

(1) Simple equations, or equations of 
the first degree, are those which involve 
the first power of an unknown quantity. 
Thus, 


w= @ 
a+tb=et+d 


e=a(p+7+*) 


are forms of simple equations, because the 
unknown quantity represented by a, does 
not appear except in the first power; @. @,, 
there are no powers of w of the type a 
and no roots of 2, of the type Vz. 

(2) Quadratic equations, or equations of 
the second degree, are those in which occur 
the second power of an unknown quantity. 


fe: THE INTERPRETATION OF 


Thus 
ray 
avet+ be+te=O0 


are examples of quadratic equations. 

(3) Lquations of the third degree, or those 
involving third powers of an unknown 
quantity: Thus, 





Tigi t 
aw + ba? +ca+d=0 
3 | 
Slee |. age 
C ad 


are examples of equations of the third 
degree. 

Similarly, equations may be formed of 
any degree. Equations of the first and 
second may be solved by definite rules; 
many of those of the third degree may be 
solved; but equations of the fourth, or 
higher degrees, can only be solved rigor- 


MATHEMATICAL FORMULZ. 73 


ously in special cases, although arithmet- 
ical approximations to their solution can 
in all cases be obtained to any desired 
_ degree of accuracy. 

If the same algebraic operation be per- 
formed upon both sides or members of an 
equation, the equality remains unaltered, 
although the form of the equation may be 
ereatly changed. Lor example, if 


med 
then 
etb=at+b 


because a certain quantity 0, is added to 
both # and a, and if these latter are equal, 
they must remain equal when increased by 
the quantity 0. 


Again e=a it ec=a 
or (c+ 6) = (@ + a) 
or Vz —Va 


or Mx = MA 


14 TILE INTERPRETATION OF 


or — — ae 


In all these cases, the same operation is 
performed on each side of the equation. 
Many transformations may be effected by 


this process. Jor example, in the equa- 
tion 
e+az= 10, 


if we subtract the quantity a, from each 
side of the equation, we obtain 
et+a—-a=10-a4. 


On the left-hand side we now have a, 
added to a, and then a subtracted from 
the result. Consequently, the as cancel, 
or may be removed from the left-hand side, 
and the equation becomes 


42 Haat WS ee 2 


We thus see that a quantity may be shifted 
from one side of an equation to the other 


MATHEMATICAL FORMULA. 1D 


by changing its sign, because, in the orig- 
inal equation a, appeared on the left-hand 
side under the positive sign, while in the 
transformed equation it appears on the 
right-hand side, under the negative sign. 

As an example of algebraic equations 
employed in physics, we may take the fol- 
lowing : | 

If V,, be the volume of a liquid at tem- 
perature 0° C., and V,, be the volume at a 
temperature ?° C., 


V.= Vi(l t+ at + pf + yt). 


The equation states that the volume at tem- 
perature # C.,is equal to the volume at 
0° C., multiplied by a compound term; 7. ¢., 
the term within the brackets. This term is 
the sum of unity, and @ times the tempera- 
ture elevation ¢, 6 times the square of the 
temperature elevation, and » times the 
cube of the temperature elevation, 


76 THE INTERPRETATION OF 


| 


For example, if V, = 100 cubic centime- 
tres, ¢ = 10° C.; a. = 0.000002 
0.00000083889, yv = 0.00000007173, or 
a = 9.53 X 10-5 6 = 8389 x 10% 
y = 7.173 X 10~*. 

Then 


V,= 100 (1 + 10 X 2:53 X 10° Ose 
8.389 X 10-7 + 108 x 7.173 x 10°) 
100 (1 + 2.53 X 107° +:8:389 4 
await ie Wess ce ATs 8) 
100 (1 + 0.0000253 + 0.00008389 
+ 0.00007173) 
AROLOIS GPs 0.00018092) 
100 (1.00018092) 
100.018092 cubie centimetres. 


As an example of an equation involving 
an infinite series, the following may be 
considered : 


2 


7 1 1 ] <I ? ae 
eo pletgt ett ae 


MATHEMATICAL FORMULZ. Pig 


This shews that the square of the quantity 
x, Which as far as four decimal places 
is 8.1416, divided by 6, is equal to the sum 
of the reciprocals of the squares of the suc- 
cessive natural numbers carried to infinity. 
It is evident that the value of z, might be 
computed by the aid of such a series. It 
could never be determined with absolute 
accuracy because an infinite number of 
terms would be required, but it could be - 
computed to any desired number of 
decimal places. 


CHAPTER VIII. 
LOGARITHMS, 


In the equation 


the power 2, to which the base 5, is raised 
in order to be equal to 25, is called the 
logarithm of 25, to the dase 5. Similarly, ' 


Oe 


may be stated by saying that n, is the 
logarithm of the number a, to the base a; 
ad, may be any positive number, and n, 
may be any number, positive or negative, 
and integral or fractional; 2. @, a whole 
number or a fraction, or a whole number 
and a fraction. | 
8 


MATHEMATICAL FORMULA. "9 


In applied mathematics, logarithms are 
of considerable importance in performing 
calculations, as well as in theoretical dis: 
cussions. The base which is_ usually 
selected is 10, and logarithms to this base 
are called common logarithms. 

From the general law of the addition 
of exponents in forming a product, we 
know that 


10? X 10? = 10¢*® = 10°. 


On the left-hand side of this equation the 
logarithms of the two factors or quantities 
to be multiplied are, respectively, 2 and 3, 
while the logarithm of the product is their 
sum, 5. This rule is of general applica- 
tion, and may be expressed as follows: 

If we sum or add the logarithms of two 
numbers, we obtain the logarithm of their 
product. 


80 THE INTERPRETATION OF 


Thus the logarithm of 100 = 2, because 
10? = 100; the logarithm of 1,000 = 3, 
because 10? = 1,000; and the logarithm of 
100 X 1,000 = 5, because 10° X 10? = 10°. 

If we suppose that a table of logarithms 
to the base 10, is prepared for all numbers 
between 0, and infinity, not merely for 
whole numbers, but for all decimal parts 
as well, we obtain what is called a table 
of logarithms. All numbers which He be- 
tween 1 and 10, will have a logarithm 
lying between 0 and 1, because 10° = 1 
and 10° = 10. 

Similarly, all numbers lying between 10 
and 100, will have a logarithm lying be- 
tween 1 and 2, because 10! = 10, and 10? = 
100. Similarly, any number lying be- 
tween 10* and 10°*?, will have a logarithm 
lying between a and a + 1. 

If we take up a table of common log- 
arithms and look for the logarithm of a 


MATHEMATICAL FORMULA. 81 


number, say 15, we find the decimals 
1760913. This represents the decimal 
part, or mantissa, of the logarithm, and the 
characteristic, or whole number, is to be 
supplied by the student. Since 15, lies 
between 10 and 100, its logarithm hes 
between 1 and 2, or is 1 and some frac- 
tion; the decimal fraction being .1760913, 
the complete logarithm is 1.1760913, which 
may be stated symbolically as follows: 


, 1 (01-160918 — Lb: 
or, 
log,, 15 = 1.1760918. 


Here the subscript 10, denotes that 10, is 
the base employed. Strictly speaking, an 
indefinitely great number of decimal places 
in the fractional part or mantissa would 
have to be employed to obtain the abso- 
lutely true logarithm, but for all practical 
purposes it is found that seven places of 


82 THE INTERPRETATION OF 


decimals are sufficient, and in many cases 
even five places are employed for the ordi- 
nary degree of accuracy required. We 
have seen that the logarithm of 15, is 
1.17609138, but the logarithm of 150, will 
only differ from the logarithm of 15, in the 
characteristic or whole number. This 
characteristic must lie between 2 and 8, 
which are the logarithms of 100 and 1,000, 
respectively, so that the complete logarithy 
of 150 is 2.1760913. 
This may be obtained in another way 
since | 
15 = 101-1600 
and 
10 = 10! 
10 x 15 = 10! x 1 1-2460918 = 1 O@- 1760918) 


In the same way the logarithm of 1,500, 
‘or 
log 1,500 = 3.1760913, 


MATHEMATICAL FORMULA. 83 
log 15,000 = 4.1760913, 


log 1.5 0.1760913, 
log 0.15 = 1.1760913, 


or has a characteristic of — 1 


log 0.015 = 2.1760913, ete. 


Similarly, the logarithm of the number 
16, is found from the tables to be .2041200 
for the mantissa, and the characteristic 
is 1, because 16 lies between 10 and 100; 
therefore, the complete logarithm is 
1.2041200, or 16 = 1017041200, 

If now we multiply 15 by 16, or make 


— 1.1760913 4 
ee 913 x 10! Sawees 
we obtain as their product 


he 1 Q4.1760918 + 1.2041200) — 1 0 @.3802118) 


Here az, is the number whose logarithm 
9.3802113 is greater than 2 and less than 


84 THE INTERPRETATION OF 


8. Consequently, a, lies between 10° = 
100, and 10? = 1,000. Entering the table 
of logarithms for the number whose 
mantissa is .8802118, we find that this 
corresponds to 24, so that the answer 
is 240, or 


15 X 16 = 240, 


Here we have performed the computation 
without the aid of arithmetical multipli- 
cation, by the summation of logarithms. 
This rule is of general application. If we 
desire to form a product of two or more 
numbers, we find their logarithms and add 
them together; the sum is the logarithm 
of the product sought. As an example 
of the application of logarithms, we may 
consider the following: 

What is the number of feet in the cir- 
cumference of the earth considered as a 
sphere of diameter 7,918 miles of 5,280 feet 


MATHEMATICAL FORMULA. 85 


each? Here the circumference may be 


written 
O=n X 7,918 X 5,280, 


or, taking z as 3.1416, 
CO = 3.1416 X 7,918 X 5,280. 


If we obtain tbe logarithms of 3.1416, 
7,918 and 5,280, and add them together, 
their sum will be the logarithm of the 
quantity C, required. The logarithm of 
8.1416 will lie between 0 and 1, or will 
have a characteristic of 0. The logarithm 
of 7,918, will lie between 8 and 4, or will 
have a characteristic of 8. The logarithm 
of 5,280, will lie between 8 and 4, or will 
have a characteristic of 8. Consequently, 
we find by reference to a table of loga- 


rithms 
log 3.1416 = 0.4971509 
“7918 = 3.8986155 
“ 5280 = 3.7226339 
8.1184008. 


86 THE INTERPRETATION OF 


Consequently | 
C! = 108:118#003 


and C, lies between 108 and 10% The 
number corresponding to the mantissa 
0.1184008 is found from the tables to be 
1.31341, so that 


C= 1.81341 X 108 or 131,841,000, 


as far as six places of decimals. If we 
make the computation we find 


3.1416 X 7,918 X 5,280 = 131,340,996,864, 


so that the arithmetically computed result 
differs from the result obtained from 
‘ seven-place logarithms by less than 4 parts 
in 131 millions. 

If now it should be required to divide 16 
by 15, or to compute by logarithms 


pete 
152 


MATHEMATICAL FORMULAE. 87 


we have already seen that the logarithm of 
16, 1s 1.2041200, and of 15, is 1.1760913. 
Consequently, 


1 (12041200 


~ 7 ()ii760078 
— 1 (12081200 y¢ 4 Q— 1-1760018 
= 1 ()4.7041200 — 1.1760913) 


1 ()(.0280287) 


Here @ is a number lying between 1 and 
10, because the logarithm has a characteris- 
tic lying between 0 and 1, and the mantissa 
0.0280287 is found in the tables to corre- 
spond to the number 1.06667. The com- 
puted quotient is found to be 1.0666666 ; 
or, as this is sometimes written, 1.06. 

As a further example of the use of loga- 
rithms, consider the following problem. 
The force of gravitation between two 


homogeneous spheres of masses m, and meg, 


88 THE INTERPRETATION OF 


grammes, respectively, is expressed by the 
equation, 





where d, is the distance between the cen- 
tres of the spheres in centimetres, and 
a, the gravitation constant; a, 1s known 
to be 6.48 X 10-8, approximately. What 
then will be the attractive force be- 
tween the earth and the moon expressed 
in dynes, if the earth’s mass is 6.02 x 10” 
grammes, the moon’s mass is 7.525 X 10” 
erammes, and the mean distance between 
their centres 3.8444 Xx 10” centimetres? 
The preceding equation may, therefore, be 
written : 


6.02 * 107 KT S25e eno 
3.8444 X 10” x 3.8444 x 10” 
_ 6.48 X 6.02 X 7.525  10* 
~ §8,8444 X 3.8444 «x 10” 
6.48 6.02. % 7.695 xaos 
3.8444 x 3,8444 . 


FE = 6.48 X 10° x 





MATHEMATICAL FORMULZ. 89 


Here the logarithms of the three quantities 
in the numerator, and also of the two 
quantities in the denominator, have all a 
characteristic of zero, since the numbers 
lie between 1 and 10, Consequently, by 
reference to tables 


log 6.48 = 0.8115750 log 3.8444 = 0.5848286 

log 6.02 = 0.7795965 log 3.8444 = 0.5848286 

log 7.525 = 0.8765065 1.1696572 
2.4676780 


The numerator of the fraction is, therefore, 
10747680 While the denominator is 10170, 
Dividing the numerator by the denomi- 
nator, or subtracting the logarithm of the 
denominator from the logarithm of the 
numerator, we have 


/ 


2.4676780 
1.1696572 
1.2980208, 


90 THE INTERPRETATION OF 


which is the logarithm of 19.8619. Conse- 


quently, 
Hr = 19.8619 xX 10” 


= 1.98619 x 10” dynes. 


Another important use of logarithms is 
in involution and evolution. or example, 
suppose that it be required to find the 
cube of 5,280, corresponding to the num- 
ber of cubic feet ina cubic mile. This 
will be equivalent to solving by log- 
arithms the equation : 

Vico 2oue 
If 

5,280 = 10% 
then 

we = (10°)? = 10™. 

Here a, is the logarithm of 5,280, and 3a, is 
three times this logarithm, as indeed is 
evident from the fact that the operation of 
cubing is multiplication to the number by 


itself twice in succession, or equivalent of 


MATHEMATICAL FORMULA. 91 


adding its logarithm to itself twice in suc- 
cession. The logarithm of 5,280, will le 
between 3 and 4, and is found from a 
table to be 

3.7226339. 


Multiplying this by 3, we obtain 


11.1679017, 
or 


aaa 11.1679017 — 11 0.1679017 
aw = LOM — 19 x 1 0160017, 


and the number corresponding to the man- 
tissa, .1679017, is 


1.47198. 
Consequently, 
v= 1.47198 X 10" = 147,198,000,000. 
The true answer is 


aw = 1.47197952 X 104% = 147,197,952,000, 


92 THE INTERPRETATION OF 


In the same way, if it should be required 
to determine the 15th root of the number 
576, or to compute by logarithms, 


ices 576%, 


we obtain the logarithm of 576, and then 
divide this by 15. 


2.760422 
log @ = ee = 0,1840281. 


CO 


which mantissa corresponds to the number 
1.52767; or, 


eID 2T BT. 


as far as five places of decimals. 
As an example of the use of logarithms 


in theoretical physics, we may consider 
the following equation : 


h = (1+ 0.003662) x 1,839,300 log centimetres, 


2 


MATHEMATICAL FORMULZE. 93 


where h, is the vertical difference of eleva- 
tion in centimetres between two stations at 
which the barometric pressures of p, and 
. px, are respectively observed, ¢, being the 
temperature of the air in °C. 


Thus, if 
ee—o0.20) andy, — 360, ¢=— 15° C,, 


then 


30.25 
30 


= (1+ 0,549) x 1,839,300 x log 1.008333 
= 1.0549 X 1,839,300 x log 1.008333 
= 1.0549 X 1,839,300 X 0.0036040. 





h =(1+ 0.00866 X 15) X 1,839,300 log 


This triple product, now cleared of loga- 
rithms, may be computed either arithmet- 
ically, or by logarithms; the result will be 
found to be 


h = 6992.7 centimetres. 


All the logarithms we have hitherto con- 
sidered have been common logarithms with 


94 THE INTERPRETATION OF 


10, for the base. In mathematical theory, 
however, and frequently in applied mathe- 
inatics, another and a more natural base sug- 
gests itself. ‘This base, as far as seven deci- 
mal places, is 2.7182818, and 1s often repre- 
sented by the symbol « Logarithms to 
this base are called natwral logarithms, 
Naperian logarithms, or hyperbolic loga- 
rithms. For all practical purposes it 1s suf. 
ficient to remember that a natural logarithm 
of a number is always greater than the 
common logarithm of the same number in 
a definite ratio, which is 2.3026, approx- 
imately, or 


2.71828" * 2-0 = 10m 


If, therefore, m, is a logarithm in the com- 
mon system, 2.30262 will be the approx- 
imate logarithm in the Naperian system. 
Thus the logarithm of 10 to the base ¢, or 


MATHEMATICAL FORMULA. 95 


log, 10 = 1 x 2.3026 = 2.3026, approxi- 
mately. 

As an example of the application of 
_ hyperbolic logarithms in apphed mathe- 
matics, we may consider the following 
equation: 


W = p, % logs (+). 


Here W, is the work done by a volume of 
gas expanding at constant temperature 
from a volume of 2, to a volume 2. Sup- 
pose, for example, that a volume of 5,000 
cubic centimetres of a gas under a pres- 
sure of 2,000 grammes per square centi- 
metre, expands to a volume of 10,000 cubic 
centimetres. Then the work done in the 
process will be 


10,000 


W = 2,000 x 5,000 x log, =m 
? 


96 


| 


MATHEMATICAL FORMULA. 

10° log, 2 

107 X 2.3026 logy 2 

2.3026 X 10° logy, 2 

2.3026 xX 10° X 0.8010300 

6.93 X 10° centimetre-grammes, 


CHAPTER IX. 
TRIGONOMETRY. 


Trigonometry is the science relating to 
the measurement of angles in a plane, with 
particular reference to triangles, and also 
to figures bounded by straight lines. An 
angle is measured by the amount of open- 
ing between two straight lines meeting at a 
point. In all practical applications, when 
this opening completes one revolution, it is 
divided into 860 equal parts called degrees, 
which in their turn are each divided into 60 
equal parts called mnutes, and each of these 
into 60 equal parts called seconds. Con- 
sequently, a complete revolution is divided 
into 1,296,000 seconds, or 21,600 minutes, 


or 3860 degrees. Almost all trigonomet- 
97 


98 THE INTERPRETATION OF 


rical tables in practical use refer to these 
angular units. For theoretical purposes it 
is found useful to employ a different unit 
called a radian. Thus in Fig. 2, suppose a 


fixed line OA, and a movable line OD, 


Fie. 2. 


capable of rotating about the fixed point 
QO, inthe plane AOL. Then, if the point 
B, traces out an are AB, the length of the 
arc AJ, is the measure of the angle a, 
included between OA and OL. If the 
angle a, be such that the are AS, is equal 
in length to the moving radius, or vadius 
vector OB, then the angle a, is 1 radian. 


MATHEMATICAL FORMUL®. 99 


This angle expressed in degrees is 57° 17’ 
44.8", approximately. Ifthe length of the 
radius vector is taken as unity, then the 
_magnitude of the angle a, will be the 
length of the are AL. It will be evident 
that if we trace out a complete revolution 
or circle with the radius vector, the length 
of this circumference will be 27 units, 
and, therefore, the angle corresponding to 
a complete revolution is 360°, or 2z 
radians. Similarly, half a revolution is 
180°, or z radians; a quarter of a revolu- 


tion is a right angle of 90°, or a radians ; 
and, generally, an angle of m degrees is an 


angle of 0 x m radians. 

An angle may extend beyond 360°. 
Thus two complete revolutions represent 
720°, or 4 radians, and 8 complete revolu- 


tions 1,080°, or 67 radians, and so on. In 


100 Titi INTERPRETATION OF 


most practical applications, however, 
angles of less than 360° are considered, 
and, in the majority of cases, angles less 
than 90°. 

An angle is considered positive when the 
direction of rotation of the radius vector is 
counter-clockwise; that is, the opposite direc- 
tion to the rotation of the hands of a clock, 
when viewed from in front of the clock. 
If the rotation be made in the opposite 
direction, or clockwise, the angle is nega- 


tive. Thus the angle AOS, Fig. 3, may 


e v7 
be regarded either as — 45°, or — ee 
(7 
dians; or, as + 315°, or + or radians. 


When the angle considered has the radius 
vector between A and -A,, Fig. 3, it is said 
to lie in the first quadrant. When the 
radius vector lies between A, and A,, it is 
said to lie in the second quadrant. When 


MATHEMATICAL FoRMULA. 101 
between A, and A,, in the third quadrant, 
and when between A, and A, in the 


fourth quadrant. 
Besides the magnitude of an angle itself, 





as determined in degrees or in radians, 
there are several important magnitudes 
connected with it which are called the 
trigonometrical functions, and which must 
be memorized since they constantly occur 


102 THE INTERPRETATION OF 


in all trigonometrical writings. These 
trigonometrical functions or ratios, some 
times called the circular functions, are as 
follows : 


The sine ; the cosine. 
The tangent ; the cotangent. 
The secant, and the cosecant. 


Occasionally the versed sine and coversed 
sine are added, but they are rarely used. 
In Fig. 4, the angle a, is represented as 
being contained between the fixed line 
OA, called the znztial line, and the radius 
vector OL. If we let fall a perpendicular 
DBC, from the point SL, on to the initial line 
we obtain a right-angled triangle OLC 
Then the ratio of the length of the per- 
pendicular LC; to the length of the hypoth- 


enuse, or radius vector OJ, or the frac- 


tion BE. is called the szne of the angle a, 


OL 


MATHEMATICAL FORMULE. 103 


and is writtenin abbreviation sin a. Conse: 


i per licular 
sequently, sin a = BC _ perpendicular 


OB hypothenuse ° 


o--—~------—----y@ 





Fie. 4. 
If the hypothenuse or radius vector be 
chosen of unit length, then the length of 
the perpendicular is the sine of the angle 
a,or sn a = BC. Thus, if a = 60° and 
OB, is unity, it will be found that the 
length of BC, is approximately, 0.866, so 
that sin 60° = 0.866, as far as three decimal 
places. By reference to trigonometrical 


104 TIE INTERPRETATION OF 


tables it will be found that sin 60° = 

0.8660254, as far as seven decimal places. 
The ratio of the length of the base OC, 

to the hypothenuse or radius vector OB, is 


called the cosine of the angle a, so that 


0 IC = cosine a, which is abbreviated to 


OL 
cosa. If OL, be unity this ratio becomes 
simply OC, and OC'= cosa. Foranangle 
of 60° as shown, the length of OC, will be 
found to be exactly half that of OJ, or 


i 
cos 60° Sr eee 0.50. 


The eit of the length of the perpen- 
dicular BC, to the length of the base OC, 


or the fraction ae is called the tangent 


of the angle a, and is abbreviated tan a. 


, BG 
Thus, for an angle of 60° the ratio OO 


will be found to be 1.732, approximately, 
and by reference to trigonometrical tables 


MATHEMATICAL FORMUL. 105 


the tangent of 60° is given as 1.7320508, as 
far as seven places of decimals. 

Similarly, the ratio of the length of the 
base OC, to the length of the perpendic- 
OC 
BO? 


the reciprocal of the fraction representing 


wlar LC, or the fraction (which is 


the tangent) is called the cotangent of the 
angle a, and is abbreviated cot a. Conse- 


OU 
quently, cot a =—,, the value of the co- 


BO? 
tangent of an angle, will evidently be the 
reciprocal of the tangent, so that, since we 
have seen that the tangent of 60°, or tan 
60° = 1.732, we know that the cotangent of 


| 1 
the same angle, or cot 60° = Toa ae 


0.5773503, approximately, as far as seven 
places of decimals. 

The ratio of the length of the radius 
vector, or hypothenuse LV, to the length of 


106 THE INTERPRETATION OF 


the base OC, or the fraction represented 





by ea is called the secant of the angle 


a, and is abbreviated see a. Consequently, 
BC 
OC” 


be written sec a = 


sec a = 


With OS, unity, this may 


1 
aE 
evidently the reciprocal of the cosine of 
1 
cos a’ 
since we have seen that the cosine of 60° = 


0.5, the secant of 60°, or sec 60° = = 
= 50, 

The ratio of the length of the radius 
vector or hypothenuse OJ, to the length 


This ratio is 








the same angle, or sec a = and 


of the perpendicular BC, or the fraction 


represented by oan is called the cosecant 


of the angle a, and is abbreviated cosec a. 


Consequently, cosec a = =~. With 


MATHEMATICAL FORMULA. 107 


OB, unity, this may be written cosec a = 


ia This ratio is evidently the recip- 


BOC 


rocal of the sine of the same angle or 


and, since we have seen 





cosec a = 
SIN @ 


that the sine of 60° is, approximately, 


(0.866, the cosecant of 60°, or cosec 60° = . 
1 
0.866 

imal places. 
It is important to remember that 


= 1.1547005, as far as seven dec- 








iL 
Cot we = == 
tan @ 
I 
sec a= 
COS @ 
1 
coset a = — ° 
sin @ 


If we study the increase of an angle in 
the first quadrant, it will be seen that the 
length of the perpendicular LC, which 


108 THE INTERPRETATION OF 


with unit radius is the sine, increases 
steadily, although not proportionately, from 
0, when the angle is zero, to unity at 90°, 
when /C, coincides with the radius vector. 
The sine of an angle may have any 
numerical value between 0 and 1, but can- 
not exceed unity. The same is true as 
regards the cosine, but in the case of the 
tangent its value commences at zero, when 
the angle is zero, but increases indefinitely 
as the angle approaches 90°, because, while 
the perpendicular “£C’ then approaches 
unity, the base OC’ becomes indefinitely 
small and, therefore, the ratio of the per- 
pendicular to the base becomes indefinitely 
great. This is expressed by saying that 
tan 90° = infinity; or, as it is usually 
written, tan 90 = o. 

Similarly, the cotangent of an angle 
commences with an indefinitely great 
value and diminishes as the angle is in- 


MATHEMATICAL FORMULA. 109 


creased, until it becomes 0 when a = 90°. 
The cosecant commences at infinity when 
a = 0, and diminishes to unity when a 
-= 90°. The secant commences at unity 
when a = 0 and increases to infinity 
when a = 90°. 

As we increase the angle beyond the 
first quadrant as represented in Fig. 4, we 
cause the values of the trigonometrical 
functions to repeat themselves cyclically. 
Thus, in Fig. 5, the length of the perpen- 
dicular let fall from the end of the radius 
vector of unit length upon the initial line 
is always the sine of the corresponding 
angle. Thus, when OS, reaches 60° at B,, 
L, CU; = 0.866, approximately, or sin 60° = 
0.866. When, for example, OD reaches 
150° at 6, the length B, C, is 0.5, and 
sin 150° = 0.5. When OB, passes into 
the third quadrant, the perpendicular CB, 
is below the initial line, or is negative in 


110 THE INTERPRETATION OF 


value; consequently, at such a point, for ex- 
ample as OS, corresponding to a = 240°, 
B, C; = —0.866 and sin 240° = —0.866. 






oO 





¥ a 
Fie. 5. 


Again, when the angle passes through the 
fourth quadrant the perpendicular LC, is 
still below the initial line, and is, there- 


MATHEMATICAL FORMULA. ria fs 


fore, still negative, so that at the position 
OB,, corresponding to a = 350°, or —10°, 
the length L, C,, is 0.1736482, approxi- 
mately, and sin 350° = sin —10° = 
—0.1736482 as far as seven decimal places. 
The sine of the angle is, therefore, positive 
in the first two quadrants, and is negative 
in the third and fourth quadrants. 

The cosine of an angle, when described 
with unit radius vector, is equal to the 
length of the base OC. In the first quad- 
rant this base is always positive. ‘Thus, in 
Fie. 5, cos 60° = OC, = 0.5. In the sec- 
ond quadrant the base OC, lies on the 
left-hand side of the origin O, and is, 
therefore, reckoned as negative. Conse- 
quently, the cosine of all angles in the 
second quadrant is negative. At the posi- 
tion indicated, OL,, the cosine OC, 1s, 
approximately, —0.866, and cos 150° = 
—0.866, approximately. Similarly, in the 


be TILE INTERPRETATION OF 


third quadrant, the base OC, still lies to 
the left-hand side of the origin, and is 
negative, so that the cosine of the angle 
240° is OC; = —0.5. In the fourth quad- 
rant, the base falls on the right-hand side 
of the ongin and is positive. Conse- 
quently, cos 850° = cos —10° = OG, = 
0.9848079, approximately. The cosines 
of angles in the first and fourth quadrants 
are, therefore, positive, and in the second 
and third quadrants, negative. 

In the same way the tangent, cotangent, 
secant and cosecant could be followed out 
through the different quadrants, always 
remembering that a base is negative when 
drawn on the left-hand side of the origin, 
and a perpendicular is negative when 
drawn below the origin, while the radius 
vector remains positive throughout the 
revolution. It is sufficient to observe, 
however, that the tangent of the angle is 


MATHEMATICAL FORMULA. i og BS’ 


the ratio of the sine to the cosine. Thus, 


m Hig, 5, 


sin a = BC 
OB 

eh OG 
we 


ee BCs BO) OB 
oe” (OC OB. OC 











Consequently, if the sine and cosine of an 
angle are found, we obtain the tangent by 
dividing the latter into the former, while 
the cosecant, secant and cotangent are the 
reciprocals of the sine, cosine and tangent, 
respectively. Thus, we have seen that 
sin 240° = — 0.866, approximately, and 


114 THE INTERPRETATION OF 





that the cos 240° = — 0.5. Therefore, 
— 0.866 
tan 240° = = + 1.7382. 
east) 
1 
) 9AI)O — Ra “ 
Then cosec 240° = Sn 1.1547 
E af 
sec 240 mary 
cot 240° = Sue = + 0.57735 
ee yey ; 


The foregoing rules will enable the stu- 
dent to understand how the trigonometri- 
cal ratios or functions are defined or 
measured for all angles. The science of 
trigonometry deals largely with the appli- 
cations of these ratios. There are a num- 
ber of important relations connecting the 
trigonometrical ratios. Thus, whatever 
the angle a may be, the following equation 
holds true, 
sin? a + cos*a = 1; 


or the square of the sine of the angle a, 


MATHEMATICAL FORMULZ. 115 


added to the square of the cosine of that 
angle, gives unity. 
Thus, in the case of Fig. 3, 


sin 60° = 0.866 and cos 60° = 0.5. 


Therefore, sin? 60° = (0.866)? = 0.750 
and cos* 50° = (0.5)? = 0.250. 
Therefore sin? 60° + cos? 60° = 1.000. 





From these relations it will be evident 
that if we know the sine of an angle we 
ean obtain its cosine, and from these the 
tangent and the reciprocal values, the’ 
secant, the cosecant and the cotangent. 

It is not the intention of this book to 
develop the theory of trigonometry, but 
merely to enable a student intelligently to 
comprehend or interpret the meaning of 
any trigonometrical formula. We must 
refer him to treatises on trigonometry for 
such developments as will enable him to 
solve problems in trigonometry. We take 


116 THE INTERPRETATION OF 


the following cases, however, as examples 
of trigonometrical equations: 

Suppose that a tall object, such as a flag- 
staff AL, Fig. 6, has an unknown height 
which it is desired to measure, and that 
the observer at O, on the same level as the 
bottom A of the staff, observes the magni- 





Fig. 6. 


tude of the angle AOL = a, which the staff 
subtends at hiseye. Then, if the observer 
measures the length of the base OA, which 
separates him from the staff, he is at once 
enabled to compute the height of the staff 


AB 
AB, because he knows that OA = tan a. 


~ 


MATHEMATICAL FORMULA. ala Be 


If we multiply both sides of this equation 
by OA, we have 


AB 
Giaees OA = OA tana 


or ibe A tanto. 


Consequently, the unknown height AB, is 
equal to the product of the measured base 
OA, and the tangent of the observed 
angle. ‘Thus, if OA, were 100 feet and the 
angle a, was observed to be 30° 35’, then 


AB = 100 X tan 80° 35’; 
= 100 Xx 0.5910 = 59.1 feet, 
approximately. 

The attraction of gravitation 1s known 
to be different at different points of the 
earth’s surface. Its value depends upon 
the latitude, and upon the elevation of the 
place at which the observation is made. 
The formula is usually expressed as follows: 

g = 930.6056 — 2.5028 cos 27 — 0.000008 h, dynes ; 


118 THE INTERPRETATION OF 


where g, is the gravitational force in dynes 
or C. G. S. (centimetre-gramme-second) 
units of force, 7, 1s the latitude of the 
locality, and f, is its height in centt- 
metres above the level of the sea. 

Thus, at the latitude of New York, which 
is, approximately, 40° 43’, and at an eleva- 
tion of 10 metres, or 1000 centimetres 
above the mean tide level, the gravitational 
force would be: 


g = 980.6056 — 2.5028 cos 2 (40° 43°) — 
0.000003 x 1,000; 
= 980.6056 — 2.5028 cos (81° 26) — 
0.008 ; 
= 980.6056 — 2.5028 x 0.14896 — 0.008 ; 
= 980.6056 — 0.3728 — 0.003; 
= 980.2298 dynes. 


As another example of a trigonometrical 
equation we may consider the following: 


t= ZL cos a, 


MATHEMATICAL FORMULA. 119 


Here JZ, represents the length of a degree 
of longitude on the earth’s surface measured 
at the equator, a is the latitude of any 
_ place on the earth’s surface and / the 
length of a degree of longitude at that 
place. ‘The equation states that the length 
of the degree varies as the cosine of the 
latitude, and since the cosine of an angle 
diminishes from 1 at 0°, to 0 at 90°, it is 
evident that at the pole the length of the 
degree of longitude would be indefinitely 
small, while at the equator itis, approxt- 
mately, 69.6 geographical miles. The 
length of a degree longitude at New York, 
or any place having the latitude of New 
York, is 
1 = 69.6 X cos 40° 43°; 
= 69.6 X 0.75794; 
= 52.81 miles. 

In the problem of determining the apparent 
time at sea, and, therefore, with the aid of 


120 THE INTERPRETATION OF 


the chronometer, the longitude of a ship, 
the following equation is employed : 


H SRR 
cos (=) = sin S sin (S— Z)cosec P cosec C. 


Here Z, is the angular distance of the sun’s 
centre from the zenith, or point immedi- 
ately overhead at the ship, as determined 
by the sextant; P, is the co-declination, or 
polar distance, of the sun; 2. e., the angular 
distance of the sun’s centre from the pole 
of the earth, as it would be seen by an 
observer at the centre of the earth, if the 
earth’s mass were transparent; C; is the 
co-latitude, or the latitude of the ship sub- 
tracted from 90°; 7. @, the angular dis- 
tance of the ship from the earth’s pole. S, 
is the half sum of P, Cand Z; 

aticgeit Oren 72% 

2 

or an angle equal to half the sum of the 
three angles above defined ; and 7, is the 


or S = 


MATHEMATICAL FORMULAE. 121 


hour angle, or the apparent time at the 
ship, expressed in degrees. The equation 
states that the cosine of half the hour angle 
is equal to the square root of the product 
of four quantities; the first is the sine of 
the half sum; the second is the sine of the 
half sum reduced by the angle Z, the 
third is the cosecant of the polar distance ; 
and the fourth is the cosecant of the co- 
latitude. ‘These quantities may be found 
from trigonometrical tables when the three 
angles , C’ and Z, are known. It is 
usual, however, to make the computation 
with the aid of logarithms as_ already 
described, since the multiplication of the 
four trigonometrical functions is performed 
logarithmetically by adding four loga- 
rithms, and the square root is then readily 
obtained by dividing the logarithmic sum 
by two. For this reason it is customary to 
seek in the logarithmic tables, not the 


122 TIE INTERPRETATION OF 


simple sine or cosecant, but the logarithm of 
that sine or cosecant. In practice this opera- 
tion 1s carried out by the mere addition of 
logarithms by the navigator, without the ne- 
cessity of writing down the above equation. 
_ Thus, suppose aship, at 52° 12° 42" north 
latitude, observes, with a sextant, that the 
altitude of the sun’s centre above the 
horizon is 89°5'28"; the declination of 
the sun, or its angular distance above the 
earth’s equator, as it might be observed 
from the centre of the earth, obtamed by 
tables, being 15° 8'10" north. Required 
the apparent time at the ship the moment 
the observation was made. 
Here Z, the zenith distance 
of the sun’s centre, is 
90° — (39° 5’ 28") = 50% 54° 82": 
P, the polar distance of the 
sun’s centre, 1s 
90° — (15° 8’ 10°) = (4 es 


MATHEMATICAL FORMULZ. 123 


C, the co-latitude of the ship, 
is 
90° — (52° 12’ 49") = 37° 47’ 18" 
The sum of these angles is 163° 33' 40" 
and S, the half sum, is 81° 46’ 50”. 


Since S = 81° 46’ 50”, sin S = 0.989726. 
“« S—Z= 81° 46’ 50" — (50° 5432") = 
B0° 52’ 18’, sin (S — Z) = 0.51312. 
“ P = 74° 51' 50’, cosee P = 1.086. 
“ C= 87° 47 18’, cosec C = 1.682. 


These values are found by reference to 
trigonometrical tables. 
The equation becomes, therefore, 


H ane OEY pee gi hf eA) SSeS ee Ee Se hes I , oie eee 
cos (5) = 4/ 0.989726 x 0.51312 X 1.036 X 1.632. 


= 0.9266 
= cos 22° 5', approximately. 
“. a = 99° 5’, 


or HH = 44° 10’. 


124 THE INTERPRETATION OF 


Every 15° represents one hour of time, 
and every 15’ represents 4 minutes of time. 

Consequently, /Z = 2 hours, 56 minutes, 
40 seconds, approximately. 

Arithmetical computation in this case 
is seen to be lengthy and tedious, but it 
is greatly facilitated by employing log- 
arithms as follows: 


Since S’= 81° 46’ 50" the 
logarithm of the sine of S, by 


tables, is = 9.99552 
(S — Z) = 30° 52' 18’, log sin 
(S— Z)orLsin(S—Z) = 9.71020 
P = 74° 51’ 50", log cosec P, 
or L cosee P = 10.01535 
C’= 87° 47' 18", log cosee C; 
or LZ cosec C = 1021273 


The logarithm of the product 39.93380. 
Dividing this sum by 2, to | 
extract the square root, = 19.96690. 


MATHEMATICAL FORMULZ. 125 


This corresponds to the logarithm of 


~/ 


the cosine of 22° 5 

.. = 22° 5’ 00’ or H = 44° 10’ 00’, 
and this angle expressed in time, allowing 
15° to the hour, 15’ to the minute, and 15” 
to the second, gives 2h."56m. 40s. 

It will be observed that in trigonomet- 
rical tables, 10 is always added to the 
characteristic of a logarithm, so as to 
avoid the use of negative characteristics. 
The logarithm 10.01535, therefore, really 
represents 0.01535, while 9.01535 repre- 
sents 1.01535, and so on. 

The area of a triangle two of whose 
sides have lengths @ and c, while their 
included angle is #, is expressed by the 
equation 


A = 5 aesin B 


This equation states that the area is 
equal to half the product of the two sides 


126 THE INTERPRETATION OF 


into the sine of the included angle. Thus, 
if @ = 50 feet, c = 60 feet and JS, the 
included angle = 45°, sin L = 0.7071068, 
and 

A =4-x 50 x 60 x 0,70710Gee 
1,500 x 0.7071068: 
= 1,060.66 square feet. 


It is sometimes convenient to employ in 
trigonometrical equations what is called 
inverse notation, with the meaning of 
which, therefore, the student should be 
familiar. 

The ordinary expressions sin a, cos a, 
tan a, sec a, cosec a, cot a, refer to the 
trigonometrical ratios of the angle a. The 
inverse expressions sin~' a, cos~’ a, tan! a, 
sec”’ a, cosec! a, and cot~’ a express the 
angles of which the sine, cosine, tangent, 
secant, cosecant, and cotangent are respec- 
tively equal to the quantity a. 


MATHEMATICAL FORMUL. LO 


Thus tan-* 1 is an angle whose tangent 
is-1, or is 45°. 
cos“ 1 is an angle whose cosine is 
L, ons. 
sin-' (4) is an angle whose sine is $ 
or 30°. 
The equation 


tan7! 9=—ea 


means that a, is an angle whose tangent is 
2,or 63° 26’. This will be evident on tak- 
ing the tangent of both sides of the 


equation : 
tan (tan 2) = tan a, 


Here the two operations tan, and tan—', are 
inverse to each other or cancel, leaving 


= tan a 


from which, by trigonometrical tables, 
a = 63° 26, approximately. 


CHAPTER X. 


HYPERBOLIC TRIGONOMETRICAL FUNCTIONS. 


We have seen that plane trigonometry 
deals with the trigonometrical or circular 
functions: 7. é., those in which the radius 
vector describes a circle, as shown in Figs. 
2, 3,4 and 5. There are, however, certain 
mathematical applications in which the 
functions are hyperbolic; 7. e, those in 
which the radius vector describes a hyper- 
bola. In Fig. 7, the curve A LCis part of 
a rectangular hyperbola ; 7%. e., part of a sec- 
tion of a right cone whose vertical angle is 
90°, when cut by a plane parallel to the 
axis of the cone. If the distance OA, 
which is called the semd-axis of the curve, 


is taken as unity, and the radius vector OB, 
128 


MATHEMATICAL FORMULA. 129 


turning around the fixed point O, moves 
over the hyperbola ABC; then twice the 
area comprised between OA, OB, and the 
curve ABC, is the hyperbolic angle de- 





Fig. 7. 


scribed by the radius vector. This hyper- 
bolic angle is not equal to the angle A OB, 
between the semi-axis OA and the radius 
vector OF. Thus, the shaded area 
marked OAS, in the figure is, approxi- 


130 THE INTERPRETATION OF 


mately, 0.44 square inches, if the base or 
semi-axis (A, is one inch. Consequently, 
the hyperbolic angle traced out in this 
case by the radius vector OB, is 2 X 0.44 = 
0.88, approximately. As the radius vector 
advances along the hyperbola ALC, the 
shaded area rapidly increases, and with 
it the hyperbolic angle, but even when the 
radius vector is infinitely long, or extends 
over an infinite length of the curve ADC, 
the circular angle between OA and OB, 
cannot exceed the angle AOJF, since OF, is 
what is called the asymptote of the curve, 
or the line which continually approaches 
the curve ALC, but only meets it at 
infinity. 

If we let fall a perpendicular BD, from 
the end 5, of the radius vector upon the 
base line OY, the length LD, is called 
the hyperbolic sine of the hyperbolic angle. 
Thus, in the case represented in Fig. 7, 


MATHEMATICAL FORMULA. 131 


BD =1,if OA = 1. Consequently, the 
hyperbolic sine of the hyperbolic angle 
0.88 is 1. This is represented sym- 
bolically by the equation: 


sinh 0.88 = 1, 


which may be read “the h-sine of 0.88 is 
unity.” 

Similarly, the length OD, from the 
origin to the foot of the perpendicular LY, 
is the hyperbolic cosine of the hyperbolic 
angle; or 


OD = cosh a, 


which is read the length OP (when OA 
= 1) isthe h-cosine of the hyperbolic 
angle a In the case represented in Fig. 


(, a = 0.88, approximately, and OD = 
1.41, approximately, Consequently, 


1.41 = cosh 0.88. 


From these two quantities all the remain- 


13D THE INTERPRETATION OF 


ing hyperbolic functions may be immedi: 
ately deduced. | 





sinh 
Thus, tant w= re 
: cosh a 
3. 
seth “aw. = see 
% cosh a’ 
1 
cosech a = ———= 
sinh a’ 
ecoth a = oe 
tanh a 


These relations are precisely similar to 
those which apply in the case of the cir- 
cular functions, that is to say, the tangent 
is the ratio of the sine to the cosine, while 
the secant, cosecant and cotangent are 
respectively reciprocals of the cosine, sine 
and tangent. In the case of Fig. 7, 

tanh 0.88 = a5 = 0.707, approximately ; 


il 


sech 0.88 1.414 


= 0.707, approximately ; 


MATHEMATICAL FORMULA. 133 


cosech 0.88 = = 1.0, approximately ; 


tek] 


coth 0.88 = O70% 1.414, approximately. 


As the radius vector advances, the 
hyperbolic sine and cosine both increase 
indefinitely, while their ratio tanh a, 
approximates more and more nearly to 
unity. 

Tables are published of the numerical 
values of the hyperbolic functions, and 
by means of these tables it will always 
be possible to apply numerically such 
formule as appear in technical works. 

As an example of the use of hyperbolic 
functions we may consider the following 
formula, which gives the deflection at the 
end of a horizontal beam of uniform cross- 
section built into a solid wall or support 
at one end, and having a weight or load 


134 MATHEMATICAL FORMULA. 


FP, attached to the free end, accompanied 
by a horizontal tension Q. 
Pate ap (7 _. tanh at) 
d n 
where /, is the length of the beam, 
n=4/ @_ 
WME 
where /, is a co-efficient of elasticity, Z, the 
moment of inertia of the section. ‘Then 
the formula states that the deflection at 
the free end of the beam is equal to the 
quotient of P by Q, into a compound term 
within the bracket, the first term of which 
is the length of the beam, and the second, 
which is negative, is one nth of the hyper- 
bolic tangent of the product of the length 
and ». It, for example, n/ = 2.0 say, then 
a table of hyperbolic tangents would show 
that tanh n/ = tanh 2 = 0.964, approxi- 
mately. 


CHAPTER XI. 
DIFFERENTIAL CALCULUS. 


Suppose an observer is carried on 
a railway train and that he wishes to 
determine by observations the speed at 
which the train is moving from time to 
time. It is, of course, possible that he 
might connect a centrifugal indicator with 
the car axle and cause the instrument to 
automatically indicate or record the speed 
from moment to moment, and such instru- 
ments are actually used for the purpose, 
but without such an instrument he would 
be obliged to make an observation of the 
distance through which the train ran in a 
given time taken from a_ stop-watch. 


Thus, if in one minute by the watch the 
135 


136 THE INTERPRETATION OF 


train ran exactly three-quarters of a 
mile, then the speed of the train is evi- 
dently 3/4ths of a mile per minute, or 45 
miles per hour. This is not stating that 
the train actually maintained 45 miles per 
hour throughout the minute under obser- 
vation, because the speed might actually 
have been, say 50 miles an hour during 
some portion of the minute, and, perhaps, 
40 miles an hour at some other portion. 
It is only a statement that the mean speed 
during the minute was 45 miles per hour. 
If, however, the observer could measure 
10 seconds of time accurately, by a stop- 
watch, and observed that the train ran 
650 feet in this time, then the speed of 
the train during 10 seconds would be an 
average of 65 feet per second; or, 65 x 
225,000: 
5280 
= 42.61 miles per hour. In this case the 


3600 = 225,000 feet per hour = 


MATHEMATICAL FORMULZE. 137 


computed speed would more nearly repre- 
sent the actual speed of the train at the 
time of observation, because during 10 
seconds there would be less time for the 
speed to vary than during 60 seconds, 
but still the observer could not say posi- 
tively that the speed did not vary during 
10 seconds, and he would be still some- 
what uncertain of the exact speed when 
he commenced the observation. Now 
supposing that it was possible for him to | 
make a measurement during a single 
second, however difficult this might be in 
practice, then if the distance run-through by 
the train was exactly 66 feet in this second, 
the speed would be 66 feet per second, or 
45 miles per hour. The observer would 
now be satisfied that he had determined 
the speed of the train at the moment when 
the observation was made with a greater 
degree of accuracy, because there was so 


138 THE INTERPRETATION OF 


little time for variation to take place. 
But if the train had its brakes applied at 
the instant when the second of time was 
observed, it is quite possible that its speed 
would be rapidly falling, and that the 
speed at the commencement of the second 
would be appreciably greater than at the 
end of the second, so that even here, in the 
case of a rapidly retarded train, the time 
of observation could be too great. Reason- 
ing, however, in this way we might assume 
that supposing it were possible to make an 
observation of the distance run through by 
the train in an exceedingly small interval 
of time which we may represent by 44, 
(pronounced Delta 7¢), say the 1/1000th 
part of a second, the corresponding short 
distance which we may represent by 4s, 
(pronounced Delta s) would enable the 
velocity of the train to be computed from 


the quotient ore with a very high degree 


MATHEMATICAL FORMULA. 139 


of accuracy; and that, theoretically, if ¢, 
were made indefinitely small, as repre- 
sented symbolically by d¢, the correspond- 
ing vanishingly small distance ds, would 
enable the true velocity to be computed 
from the equation 
os 

dt i 
In this equation ds, taken by itself, has no 
meaning, because it is the space run 
through by the train in an indefinitely 
small interval of time; and so too, df, 
taken by itself, has no meaning, because it 
stands for an indefinitely small interval of 


time, but the fraction a can be consid- 


ered as the limit of = as ¢, is reduced 


indefinitely, standing for a_ perfectly 
definite notion ; namely, the instantaneous 
velocity of the train when the time chosen 


140 THE INTERPRETATION OF 


for observing it is so extremely short that 
there is no possibility of error due to 
variation of speed in that time. 

In the language of the differential cal- 
culus ds, is the differential of the space 
run through by the train; dé, is the differ- 
ential of the time during which the obser- 


vation of distance 1s measured, and 


“ 
the differential coefficient of the space s, 
run through with respect to the time t. A 
differential coefficient has, therefore, 
always the meaning of a ratio between 
the variation of two quantities carried 
to a limit. Thus the differential coeft- 


cient , 1s the limit of the ratio = when 
At, is made indefinitely small. 

The object of the differential calculus is 
to determine from a known relation be- 


tween two connected quantities the rate 


MATHEMATICAL FORMULZ. 141 


at which one varies when a variation 1s 
given to the other. Suppose, for example, 
that we consider the case of a stone falling 
to the ground from a given elevation. 
We know by observation, in connection 
with the theory of the subject, that the 
Jaw which expresses the distance through 
which the stone will fall in a given time 
¢ seconds, neglecting air-friction, 1s 


coi 596 feet, 


where g, is the constant representing the 
earth’s gravitational acceleration, or 32.2 
feet per-second-per-second, approximately. 
Consequently, the formula becomes 


Salo le feet, 
approximately. 
Having given this relation between the 
time ¢, and the distance fallen through s, 
the velocity at any instant is evidently 


142 THE INTERPRETATION OF 


defined in some way by this relation. We 
might imagine that the observation was 
made at any particular instant, say ¢ = 3, 
or three seconds after the stone com- 
menced to fall. At this time s = 16.1 


xX 3 


= "16.1 K29° S144 Steer 


By taking a small interval of time, say the 
one-tenth part of a second, thereby mak- 
ing ¢ = 3.1, we find that the space fallen 
through up to this time is 


s = 16.1 x (8.1)? = 154.72 feet, 


so that, in the interval of time 0.1 second, 
the space fallen through will have been 
154.72 — 144.9 


= 9.82 feet 


or 4s = 9.82 feet, and 4¢, the interval 
of time, 0.1 second; so that the velocity, 


MATHEMATICAL FORMULA. 148 


as gauged from this interval, will be 
4s 9.82 
ea 0,1 
But it will be evident that this is only a 
mean velocity during the time 4¢, since 


= 98.2 feet per second. 


the stone is constantly accelerated, and it 
will be somewhat in excess of the actual 
velocity of the stone at the moment ¢ = 3. 

The differential calculus supplies rules 
for determining the instantaneous velocity, 


or, in this case, the limit of > when 4, 


instead of being 0.1 second, is reduced 
indefinitely, and which, therefore, enable 


the instantaneous velocity V = to be 


ds 

dt’ 

determined accurately. It might be 

shown by rules of the differential calculus 

that the true velocity is gt, or 32.2 4, so 
ds 


= 82. 
that a 32.2 ¢, 


144 THE INTERPRETATION OF 


so that the velocity of the stone at the end 
of the third second, or when ¢ = 8, is 
2 = 96.6 feet per second, 
t 

and we should find that the result ob- 
tained above, of 98.2 feet-per-second grad- 
ually approached, the instantaneous 
velocity 96.6, as the interval of time 4¢, 
was reduced from 0,1 second to a vanish- 
ingly small quantity. 

In the case of the moving train, the 
differential calculus would be of no assist- 
ance to the observer, unless he knew the 
law which connects the distance traversed 
by the train with respect to time. We 
have only alluded to the case of a mov- 
ing train in order to give a conception of 
the limiting rate which is so constantly 
dealt with in the theory of the differential 
calculus, 


MATHEMATICAL FORMULA. 145 
Whenever a differential coefficient such 


a -/_,.is met with in a formula, it is to be 


Bs 


regarded as an actual quotient or fraction ; 
namely, the limit which the fraction a 
8 


assumes when 4s, is indefinitely reduced. 

It is always to be remembered that ds, 
in the differential calculus is not the prod- 
uct of the quantity d, into a quantity s, as 
it would be in ordinary algebra, but an 
abbreviation for “ differential of s.” 

As another example, consider the case 
of a reservoir of water, discharging 
through a pipe, and let g, be the quantity 
of water in the reservoir, and ¢, the time. 
Then, during discharge, there will be a 
certain flow of water through the pipe. 
If g, be expressed in cubic centimetres 
and ¢, in seconds, we may ascertain that 
the rate of discharge in a given time Jt, 


146 THE INTERPRETATION OF 


say 10 seconds, amounts to 4g = 15,000 
cubic centimetres. Then the average floy 


of liquid through the pipe will be 





Aq _ 15,000 


At Timer 


cubic centimetres per second. In order to 
obtain the true instantaneous value of the 
flow, it would be necessary to consider 
the interval of time 4¢, in which the 
measurement was made, indefinitely re- 
duced, as represented by the symbol di, 
and the corresponding .quantity of escape 


in that time dg, so that the instantaneour 
flow will be 


a 

She ae 
In practice the differential calculus could’ 
not here be employed, unless the relation 


between the quantity of water in the 
reservoir and the time was known, but the 


MATHEMATICAL FORMULA. 147 
dy 
dt? 


namely, the instantaneous rate of change 


conception afforded by the symbol 


in the quantity with respect to time, or 
the instantaneous flow, would be perfectly 
definite, whether it could be employed 
practically or not. 

Again, we may consider the quantity of 
heat g, which must be given to a gramme 
of a substance to raise its temperature 6°C. 


26’ 
js the mean thermal capacity of the sub- 


If we take the quotient 47 this quotient 


stance. Thus, if we take 46, as unity or 
as 1° C., then 4Q, is the amount of 
heat which must be given to a gramme of 
the substance to raise its temperature 
1°C. This quantity of heat is generally 
not the same at different temperatures, 
so that the quantity 4@Q, which would 
have to be given to a substance to raise it 


148 THE INTERPRETATION OF 


from 5° to 6° C. is not the same as that 
required to raise it from say 31 to 32° C. 
Consequently, the thermal capacity varies 
with the temperature, and a strict defini- 
tion is given by the equation 





C= et or the limiting value of hs 
when the variation 46, is made indefinitely 
small. This may be regarded as_ the 
instantaneous rate of change of heat with 
respect to temperature, at any given tem- 
perature. 

It is shown in works on electricity that 
if a loop of wire be subjected to variations 
in the quantity of magnetic flux 6, which 
passes through it, either by being moved 
through magnetic flux, as by passing the 
poles of a magnet, or by having magnetic 
flux moved through it, as by moving a 
magnet past it, the electromotive force 


MATHEMATICAL FORMULA. 149 


(abbreviated E. M. F.), generated in the 
loop, or the electric pressure tending to 
cause an electric current to flow through 
it is represented by the formula 


dp 
eet 3 


where 6, is the magnetic flux, e, the 
E. M. F. expressed in C. G. S. units and /, 
is the time expressed in seconds. This 
formula means that the E. M. F. is the 
instantaneous time-rate-of-change of mag- 
netic flux through the loop. If the loop 
at a certain instant, ¢ = 10, contains, say 
1,000,000 C. G. S. units of magnetic flux, 
and if in the next half second or at ¢ = 
10.5, the magnetic flux had increased to 
1,100,000 C. GS. units, the increase would 
amount to 100,000 in the time 0.5 second, 
and the average rate of increase during the 


half second would be 


150 THE INTERPRETATION OF 


100,000 4d | 
ye = op = 200,000 


so that the average E. M. F. would be 
200,000 C.G.S. units. If the change took 
place uniformly, this would be the actual 
iE. M. F. throughout the half second, but 
if the change occurred irregularly, the 
E. M. F. might have a value at some part 
of the time greatly in excess of 200,000 
units, and, at some other part of the 
time, greatly in deficit. But at any 
instant, if we conceived that the interval 
of time 4¢, be reduced indefinitely, the 


ratio so obtained of which would 


i 
thereby become iss is the true E. M. F. at 
ad 


40 
At’ 


that instant, no matter how rapidly the 
rate-of-change in flux may be varying. 
If there be any known law connecting the 
variable 6, or flux, called the dependent 


MATHEMATICAL FORMULA. 151 


variable, with the variable ¢, or time, called 
the endependent variable, then, from this 
law, it will be possible, by the rules of the 


differential calculus, to determine how ao 


dt’ 
will vary from moment to moment, or what 
the instantaneous I. M. I. in the loop will 
be from moment to moment. Thus, if the 
flux through the loop increases steadily 
with time, so that say @ = at, then the rules 
of the differential calculus show what is 
evident on reflection, that since each second 
adds an amount of flux to the loop equal to 
a, the E. M. F. will be @ units steadily; or 


I® 
AT — i A 


As another example, consider a simple 
pendulum consisting of a fine thread at- 
tached to a small heavy particle acting as 
the bob, This pendulum may have a cer- 


152 THE INTERPRETATION OF 


tain length / cms., and it will have a cer. 
tain periodic time, or time of one complete 
double swing, Z’seconds. Then, if we vary 
the length /, in any manner, and study the 
effect of this variation upon the periodic 
time, we make /, the independent variable or 
the variable whose change is arbitrary, and 
make 7} the dependent variable, or the vari- 
able whose changes are not arbitrary, but 
are determined by the changes inf The 
law connecting the period Z} with the 
length 7 is expressed algebraically as 


L = ony/ seconds, 


where z, is the ratio 3.14159... of a cir- 


follows: 


cumference to its diameter, /, is the length 
of the pendulum in centimetres, and g, the 
accelerating force of gravity, at the loca- 
tion considered, in centimetres-per-second- 


MATHEMATICAL FORMULA. 153 


per-second; so that the periodic time is 
6.2832 multiplied by the square root of 
the length divided by the gravitational 
constant. If now we make a change in /, 
we make a corresponding change in 7} be- 
cause as we lengthen the pendulum it 
swings more slowly, but the rate of change 
in 7; with respect to /, or the rate-of-change 
of periodic time per centimetre increase in 
pendulum length, is 


AT’ 
Ale 


where 4/, is a given increase in /, and 47} 
the corresponding increase in 7! This, 
however, is only an average rate of increase 
in periodic time, for the interval 4/, con- 
sidered, and is not the true rate at that 
length. Thus, suppose g = 981 centime- 
tres-per-second-per-second, and 7 = 100 cen- 
timetres or one metre, 


154. THE INTERPRETATION OF 


100 
981? 
= 6.283 VU.1019368 ; 
6.283 xX (0.8192); 

= 2.0055 seconds. 


= 
| 


2 X 3.1416 X 


The half cycle, or single vibration, would, 
therefore, take 1.00275 seconds, so that the 
pendulum one metre long would be nearly a 
second’s pendulum. If now we increase 
the length by one centimetre or make 
Al = 1, 

T= 2 X 3.1416 x a 
6.283 »/0.10294 ; 
6.283 (0.32085) ; 


9.0159 seconds. 


[| 


Consequently, 
AT = 2.0159 — 2.0055 = 0.0104 seconds, 


and 


MATHEMATICAL FORMULA. 155 


meee 00.104.” é. 
Maes 107 0.0104 second per cen- 





timetre. 


This is the average rate of change in the 
period of the pendulum per centimetre of 
increase in its length, but the actual rate 
of change when /= 100, is this ratio when 
Al, is so far reduced that the pendulum 
does not sensibly alter in length; or, 

al’ 

dl © 
This differential coefficient of Z} with re- 
spect to /,is known, by the rules of the 
differential calculus, to be, 


Cie ae, 
db Wig 

When 7 = 100, and g = 981, this is 
nade 3.1416 


di ~ /981 x 100° 


156 THE INTERPRETATION OF 


_ 8.1416 

= 758100 

81416» 

TEP Ey: 

= 0.01003 seconds-per-centimetre. 
Consequently, the rate of increase of per- 
iodic time per centimetre of length is 
0.01003 seconds-per-centimetre, in a metre 
pendulum, whereas the approximate process 
of calculation, assuming the actual variation 
of one centimetre, gave 0.0104 seconds-per- 
centimetre. 

If a body move in time 4, seconds, 
through s, centimetres, the instantaneous 
velocity is expressed by the differential 
coefficient of the space described with re- 
spect to the time, or 


v= ds timetr d 
= 7 centimetres per second. 


But if we consider the body moving witha 
velocity v, centimetres per second, in time 


MATHEMATICAL FORMULA. 157 


t, seconds, this velocity may be increasing 
or diminishing, and its time-rate-of-increase 
is called acceleration. This instantaneous 


: : ; et Av 
time-rate-of-increase is the limit of Ti 


and becomes 


But if we substitute for v, in this equa- 
tion its equal 


we obtain 


ds 
_ Aa) 
pre ae 


centimetres-per-second-per-second, and this 
is written 
d*s 
ie dp’ 
so that 
ds 
dé 


158 THE INTERPRETATION OF 


is called the second differential coefficient 
of s, with respect to ¢, to distinguish it from 
ds 
dt’ 
which is the first differential coefficient of 
s, with respect to 7. In the same way 


d*s 


dt?’ 


which is the third differential coefficient or 
the differential coefficient of the differen- 
tial coefficient of the differential coefficient 
with respect to ¢, and so on. It would 
mean the instantaneous time rate of accel- 
eration. 

A differential equation is an equation 
which contains differentials or differential 
cofficients. We will consider such an equa- 
tion with a view of interpreting its mean- 
ing. 

In Helmholtz’s “Sensation of Tone,” 


MATHEMATICAL FORMULA. 159 


App. IX., appears the following equation 
in relation to the vibration of a tuning fork: 
2 

mM. ee = = an ye 2 + Asin ws, 
where 2, is the excursion of a heavy vibrat- 
ing point from its mean position of rest, @ 
is a coefficient of elasticity, 4, is a coeffii- 
cient of frictional opposition to motion, A, 
is the amplitude or maximum value of an 
impressed vibratory force, ¢, the time in 
seconds starting from some particular 
epoch, », a number measuring the fre- 
quency of the variations of the impressed 
force, and m, is the mass of the heavy body 
-acted upon. The meaning of these sym- 
bols is given in the text referred to. 

The quantity on the left-hand side of the 
equation is the product of the mass m, and 


de 
de’ 


160 THE INTERPRETATION OF 


the second differential coefficient of the ex- 
cursion 2, with respect to time. The first 
differential coeificient, 

dx 

dt 
would represent the instantaneous velocity 
of the body at the time considered. The 
second differential coefficient, 

ad*e 

dt? 
is the instantaneous rate-of-change of ve- 
locity, or the instantaneous acceleration. 
The right-hand side of the equation has 
three terms. The first is the product of 
the excursion @ and the coefficient of elastic- 
ity — a. This is a force tending to restore 
the body to its initial condition of rest, and 
increases as the excursion increases. The 
second term is the product of the coefi- 
cient — 0°, and the instantaneous velocity 


MATHEMATICAL FORMULA. 161 


i) 
dt 
This is a force of friction tending to op- 
pose the motion, and increasing directly 
with the velocity. The third term is the 
impressed vibratory force, acting on the 
body, and is the product of the constant A, 
and the trigonometrical expression sin (72). 
As #, increases, the angle represented by nt, 
increases proportionately, and passes suc- 
cessively at a definite rate through all four 
quadrants. The sine of this angle, repre- 
sented by sin(vt) will, therefore, pass 
successively through 0, + 1, 0, — 1, 0, +1, 
ete., and all intermediate values. Conse- 
quently, the third term will pass through 
the values 0.4, 0 — A, 0, + A, etc; and 
all intermediate values, which 1s equivalent 
to the statement that the impressed force 
alternates at a definite rate between the 
values + A and — A, according to a sine 


162 THE INTERPRETATION OF 


law. The equation, therefores, states that 
at any and every instant the force of iner- 
tia, which is the product of the mass and 
acceleration, is equal to the sum of all the 
forces acting on the body, the first being 
the elastic force of restitution, the second 
the frictional force of retardation, and the 
third the impressed force. 

It will thus be evident that even when 
the student is not sufficiently acquainted 
with the rules of the differential calculus to 
deduce differential coefficients, or manipu- 
late them in equations, he will, neverthe- 
less, be able to interpret the meaning of 
the equations which appear in_ technical 
works employing differential calculus. 

It is shown in treatises on dynamics that 
if a rigid body be free to move about a 
fixed axis and has a moment of inertia A, 
gramme-cm’s; and is acted upon by any 
force which at time ¢ seconds, exerts a mo- 


MATHEMATICAL FORMULA. 163 


ment about the axis of d/ dyne-cms, the 
following equation holds at any and every 
instant 
ee eit 
de K 
ao . a 

Here 7g) 18 the second differential coef- 
ficient of the angle 6, described by the body 
about the axis with respect to the time ¢, or 
is the instantaneous time-rate-of increase of 
the instantaneous time-rate-of-increase of 
angle. If we call the instantaneous time-rate- 
of-increase of angle the instantaneous angu- 
lar velocity, and if we call the instantaneous 
time-rate-of-increase of angular velocity, the 
instantaneous angular acceleration, then the 
equation states that the angular acceleration 
at any instant is the quotient of the mo- 
ment of the applied force divided by the 
moment of inertia of the body. It may be 
possible to determine by the rules of the 


164 MATHEMATICAL FORMULA. 
differential calculus and so make use of the 
equation numerically, but even if the infor- 
mation is not forthcoming by which the 
equation may be practically applied, or if 
the knowledge is not available by which 
from the data of the problem the computa- 
tion may be made, still the interpretation of 
the meaning of this equation carries to the 
mind of the student a definite and perfectly 
intelligible law. 


CHAPTER XIL 
INTEGRAL CALCULUS. 


We have seen that the principal object 
of the differential calculus is to determine 
the limiting ratio of the variation in a de- 
pendent variable, with respect to an indefi- 
nitely small variation of an independent 
variable with which it is connected. The 
principal object of the integral calculus is to 
effect the inverse operation; namely, to de- 
termine from the limiting ratio of variation 
between a dependent and an independent 
variable, the fundamental relation between 
these variables. ‘Thus, from the known 
law that the distances fallen through by a 
stone in time ¢ seconds, is expressed by the 


formula 
165 


166 THE INTERPRETATION OF 
' 1 2 
Si = 397, 


where ¢, 1s the independent variable, and 
s, the dependent variable, the differential 
calculus determines the limiting relation 
between their variations, or gives the dif- 
ferential coefficient 


ae 
dt it ie g ) 

and states that the instantaneous velocity 
atany moment is equal to the product of 
the gravitational acceleration-constant g, 
and the time of descent ¢, or that for any 
very minute change in the independent 
variable amounting to dt, the correspond- 
ing change in the dependent variable is 


ds = gt.dt. 


The integral calculus would find its ap- 
plication in the inverse relation: Having 
given the observed fact that the velocity 


MATHEMATICAL FORMULA. 167 


of a falling stone is éxpressed by the 
formula, 


what will be the relation between the 
primitive variables s and ¢, corresponding 
to this condition? We shall see that any 
such inquiry naturally leads to the summa- 
tion of an indefinitely great number of in- 
definitely small terms. 

Fig. 8, shows a straight line O V, drawn 
from the origin O, to a scale such that in 
each second of time, as measured along the 
base or axis of abscisse O7' the elevation 
of the line represents the velocity of the 
falling stone, Thus, at ¢ = 1 second, the 
elevation of the point on the line above 1 
on the base, represents, to scale, 32.2 feet 
per second; at 2 seconds it is 64.4, and so 
on for all times, integral or fractional, in- 
cluded between¢ = 0 and ¢ = 2.5 = 7: 


168 THE INTERPRETATION OF 


This line, therefore, expresses graphically 
the equation 

— gt, 
where g = 382.2, as far as ¢ = 2.5 seconds. 









Joan 






Ld nes ae OLA ac Be eS 


_ a Pes a ens 
a OE 8 SRT ee 


34 
Fie. 8. t 


It will be evident that if we consider the 
moving stone at some particular instant, 
say at the time when ¢ = 1, and w = 82.2, 
the space through which the stone will 
move at this velocity in a small interval of 


MATHEMATICAL FORMULA. 169 


time 4¢, will be 4s = v4é, so that, for 





example, if 4¢ = ane second, the stone 


32,2 
1000 
time. Strictly speaking, the stone will 
have moved through more than this dis- 





will move through 4s == feet in this 


tance, because during this interval of time 
it will have been accelerated or will be 
moving faster at the end than at the begin- 
ning of the period. But if we make the 
interval of time 4¢, small enough, and theo- 
retically, if we make it indefinitely small 
or dt, the equation, 


ds = vdt, 


will become strictly true. At present, 
however, we may accept for our purposes 
the rough calculation 


As = vat 


where 4¢ = 1/4th second say, or 0.25. 


170 THE INTERPRETATION OF 


Then we may divide the whole base line 
OT of 2.5 seconds, into intervals of time Z¢ 
each equal to 0.25 second, and there will 
evidently be ten of such intervals. If 
we take the first interval, when ¢ = 0, the 
stone will evidently be just starting from 
rest, and its initial velocity will be 0. 
This may be represented by the making 
% = 0. Consequently, 


AS) = V4t = 0 X 0.25 = 0 feet. 


And the space moved through by this 
rough calculation will be 0 feet in the first 
quarter second, although we know that 
owing to taking 4¢ so long as 1/4th second, 
the result is untrue since the stone has cer- 
tainly started from rest in this time. At 
the commencement of the second interval, 
or when 


t= 0.25 v, = 0.25 X 32.2 = 8.05 feet per second. 


At this velocity, if maintained uniform, the 


MATHEMATICAL FORMULA. 171 


stone would traverse in the succeeding 
interval 4¢, a distance 


Pie 8.05. 0.25.2 2:01 90: feet: 


If we proceed in this way to determine at 
each of the ten points on the base-line O7} 
what will be the velocity at that moment, 
and what will be the distance moved 
through by the stone, assuming the velocity 
uniform during the next interval, we obtain 
the following equations: 


FEET 


At to=0 Vo 82.2 0 Aso= 0 <0.25= 0 


4:=0.25 = 8.05 As,= 8.05X0.25= 2.0125 
t2=0.50 =16.10 Asg=16.1 0.25= 4.0250 
ts=0.75 3 =24.15 As;=24.15x0.25= 6.03875 
t4=1.00 % =82.20 As,=32.2 X0.25= 8.0500 
ts=1.25 vs +» =40.25 As,—40.25x0.25=—10.0625 
ts=1.50 =48.30 As.=48.30X0.25=12.0750 
t7=1.75 7 =66.35 As,;=55.35 x0.25=14.0875 
ts=2.00 =64.40 As,=—64.400.25=16.1000 
%9=2.25 Up =72.45 Ass =72.45 x 0.25=18.1125 


At the end of the interval commenc- 
ing with ¢), the time 7’ = 2.5 seconds, 
will have been reached, and the total 
computed fall, $ = 90.5625 feet, 





Lie THE INTERPRETATION OF 


It will be evident that this result will 
necessarily be too small, owing to the 
manner in which the process has been con- 
ducted, since we have taken the initial 
velocity of the stone in each equation in 
order to obtain the distance traversed in 
the interval, thus ignoring the influence of 
acceleration during the interval. If, how- 
ever, instead of taking the interval of time 
At as 1/4th second, we reduced it one-half, 
or made it 1/8th second, each equation of 


the type 
As = vit 


would be more nearly true, and the result- 
ing sum 95.595 feet, would be also more 
nearly true, but there would be 20 equa- 
tions to sum up, instead of 10. It is evi- 
dent that as we take 4¢ smaller and 
smaller, we increase the number of equa- 
tions, but we arrive closer to the truth, 


MATHEMATICAL FORMULA. Vis 


Strictly speaking, therefore, we should 
make 4¢ indefinitely small or equal to dz, 
and sum an indefinitely great number of 
equations of the type 


ds = vdt. 


The sum of this indefinitely great number 
of eauations would give s, accurately; or 


s = sum of all terms vdt 


fanemirom ¢— 0 to\t = 2.5. This is ex- 
pressed in the language of the integral 


calculus by 
TT 
8 =| vdt, 
0 


where | stands for “the sum of all terms 


of the type.” The superscript Z} and the 
subscript 0, show that the upper and lower 
limits of the variable ¢, between which the 
summation is to be effected, are 7’ seconds 
and 0 seconds, respectively. . 

The rules and theory of the integral 


174 THE INTERPRETATION OF 
calculus show that the operation indicated 
by the integral | which it would be im- 


possible to carry out arithmetically, since 
it is impossible to write down an infinite 
number of terms, is capable of being 
directly computed without performing any 
such process. In the case considered, the 
rules of the integral calculus show that 


Gut 


s= 
2 





or, that the integral 


2 
(ea=S 
0 


9 e 





Since 


2 39.2 X 2.5 X 2.5 
7-35, 7. ee 


= 100.625 feet. 
This would be the sum arrived at were it 
possible to write down and sum up the 
indefinitely great number of terms required 
in the case considered, 


MATHEMATICAL FORMULZ. 175 


By reference to Fig. 8 it will be seen 
that the integral which we have considered 
is capable of a simple geometrical interpre- 
tation. Tor if we consider any interval of 
time 4t = 1/4th second say that following - 
the time ¢ = 1 second, the velocity at this 
time is 32.2 feet per second, represented 
by the ordinate 1v. The space traversed 
at this velocity in the succeeding quarter 
second is 


1 
As = vat = 82.2 X 7 = 8.05, 


and is evidently the area of the vertical 
strip luca, whose sides are v and Jf, re- 
spectively. From this it will be seen that 
the total space traversed in 2.5 seconds, as 
summed from ten equations, is the sum of 
the areas of the vertical strips indicated in 
Fig. 8 by the dotted lines. The true dis- 
tance traversed is, however, by the in- 
tegral calculus, 


176 THE INTERPRETATION OF 
1 
&s = 9g L e) 
In the figure, the ordinate 7’'V is equal to 


gT; and the base OZ'is equal to Z} so that 


1 


CS Leet ee units of area 


do] 


or, the area of the triangle OZ’V = 100.625. 

It is evident from the figure that as we 
reduce the length of the intervals 4¢, we 
make the vertical strips more numerous, 
and their total area will more nearly agree 
with the area of the whole triangle; and 
finally when the intervals dé are indefinitely 
small, and the steps indefinitely numerous, 
they will exactly conform at their termina- 
tions with the line Ov V, and their agere- 
gate area will exactly coincide with the 
area. of the triangle O7'V. 

Any integral may, therefore, be re- 


garded as the area of a certain geometrical 


MATHEMATICAL FORMULA. Was 


figure and to be composed of an indefi- 
nitely great number of elementary strips 
which, finally, coincide completely with 
the area. 

An important practical application of 
the process of integration is found in many 
forms of meters. Consider, for example, 
a gas meter. At any moment when the 
gas jets ina building are lighted, there will 
be a certain flow of gas into the building 
through the meter. This flow may be 
expressed by the symbol f ‘Then in any 
brief interval of time 4¢, the quantity of 
gas 4g, admitted to the building will be 


Ag = f4t 
and, if the number of gas jets remained 
unaltered and burned steadily, the quan- 
tity of gas consumed in any given number 
of hours would be 


Y = fi, 


178 THE INTERPRETATION OF 


but if a number of gas jets are turned on 
and off irregularly, or if the pressure in 
the mains varies, so that the flow is not 
uniform, then the total quantity of gas 
supplied to the house is not proportional to 
the time. At any moment, however, thz 
following equation is true: 


dq = fdt; 

dt, being an indefinitely brief interval of 
time, and dg, the corresponding minute 
quantity of flow. If we sum all the equa- 
tions which may be formed between the 
time ¢ = 7, and ¢ = 7, we obtain a per- 
fectly accurate theoretical statement for 
the quantity of gas supplied ; namely, 


Ts 
g aoon | fat. 
1, 
The meter, if properly constructed, will 


perform this integration or summation 
mechanically, and will give between the 


MATHEMATICAL FORMULA. 179 


time 7;, and the time Z}, the value of the 
integral written on the right-hand side of 
this equation. Meters of this type are 
therefore, frequently called integrating 
meters. Of course their operation does 
not require the use of the integral calculus, 
but the integral calculus enables their 
operation to be very briefly and accurately 
stated in symbolical language, and if there 
is any known law connecting the flow 
with the time, the integral is capable of 
being evaluated arithmetically. 

Fig. 9, represents the flow of gas 
through a meter, during different success- 
ive intervals. It will be seen that dur- 
ing the daytime the flow falls to zero or 
to a very small quantity. At night-time 
it rises to a maximum and varies accord- 
ing to the demand in the building. If the 
ordinates are laid off to a proper scale of 
flow, and the abscissz to a proper scale of 


180 THE INTERPRETATION OF 


time, the area of the curve between the 
ordinates Z; and 7), will give the integral 


T. 
g= | jdt. 


Ts 


which the meter mechanically registers. 






ef 


Rate of Supply 


Fie. 9. 


As another example of the meaning and 
application of the processes of the integral 
calculus, let us consider the determination 


of a height by means of barometric obser- 
vations. 


MATHEMATICAL FORMUL@. 181 


Let us suppose that at a certain station, 
say at the base of a mountain, and at some 
height 2, metres above the mean sea level, 
the barometer shows an atmospheric pres- 
sure of p, millimetres of mercury. Then, 
from the known volume of air at the 
observed temperature and at this pressure 
p, the atmosphere behaves as though it 
were a layer of air of //, centimeters in 
height having a uniform density equal to 
the density at the point of observation, 
whereas, in reality, its density continually 
diminishes as we ascend, and its real 
height is consequently far greater. The 
height /7/, is called the height of the homo- 
geneous atmosphere, or the virtual height 
to which the atmosphere would extend if its 
density remained constant at that observed. 

If now we ascend with the barometer 
through a very small distance 4h meters, 
the virtual height //, of the atmosphere 


189 THE INTERPRETATION OF 


above the original station will be reduced 
from [7 to H — 4H, while the barometric 
pressure will be reduced from p to p — 
Ap; 4p, being the small fall in the bar- 
ometric pressure due to the small differ- 
ence in height 4h. The proportional 
diminution in the virtual height of the 
atmosphere ; or, 

4H _ 4p 

Fidee 
and since the diminution in virtual height 
Aff, is the same as the change in real 


height 4h, then 


Multiplying both sides of this equation by 
Hf, we obtain 


MATHEMATICAL FORMULZ. 183 


It would be possible to call the new pres- 
sure p’, at the slightly elevated or second 
station, which is p — 4p, and repeat the 
equation for this new pressure and succes- 
sively raise the barometer say a foot at a 
time, and write down a new equation of 
this type each time: 
The elevation raised through 
i sh SN ta 
Pm 
where 7,,, 1s the pressure at any station 
in the upward series. The total height 
through which the barometer had been 
raised from the first station to the last 
would then be found by summing up all 
these equations in the following manner: 


D1 
Ah, = ee 


184. THE INTERPRETATION OF 


th, = —HP, 
Hs 


where p’, stands for the pressure at the 
last or nth station, The sum of all the 
terms on the left-hand side of these equa- 
tions 1s the total height (A’ — h) through 
which the barometer has been moved from 
the first station of elevation h, to the last 
station of which the required elevation is 
h'. The sum of the terms on the right- 
hand sides of these equations is the sum 


of all terms of the type —H = . ‘Phe 


resulting summation equation will, how- 
ever, not be quite correct, because each 
individual equation contains a small error 
due to the fact that in raising the barome- 
ter say a foot, the density of the air at 
the new elevation is somewhat less than 
that at the last preceding elevation, and 
the equation, is therefore vitiated; but if 


MATHEMATICAL FORMULA. 185 


we suppose that, instead of raising the 
barometer a foot at a time, we raised it 
through an indefinitely small distance, 
and make an indefinitely great number of 
such stages of observation, each differing 
by dh metres, the imaginary equations 
become strictly accurate, and the sum total 
becomes strictly accurate, even although it 
would be practically impossible to perform 
the experiment in this way. But by the 
rules of the integral calculus we can deter- 
mine what the sum of the indefinitely 
great number of terms on the right-hand 
side would be; for, it is expressed as the 
integral 

ep’ — Hdp 


B-2= | 
p P 


This integral is known to be 


H loges, 


186 ‘THE INTERPRETATION OF 
and the total difference of elevation, or 


(h' —h) = Hog, © 
or the total elevation, between the first 
and last station of the barometer is 
the product of //, the virtual height of 
the homogeneous atmosphere, and the 
Naperian logarithm of the ratio between 
the first and last pressures or readings of 
the barometer. Thus, if » = 750 milli- 
metres, and p’ = 375 millimetres of 
mercury, and //, the apparent height of 
the homogeneous atmosphere, 8 X 10° 
centimetres. Then 


30, 
1b 
8 X 10° log, 2 cms. 





h = 8 X 10° log, 


We may either look for the natural log- 
arithm of the number 2 in tables of 
Naperian logarithms, or we may look for 


MATHEMATICAL FORMULA. 187 


the common logarithm of 2, and multiply 
it by the constant 2.3026 so that 


h = 8 X 10° X 2.8026 X logy 2 

8 X 2.3026 x 0.3010300 x 105 
5.546 X 10° centimeters. 

5,546 X 10° metres. 

5,546 metres or 3.446 miles. 


If 


Il 


lI 


It is of course assumed either that the 
condition of the atmosphere remains uni- 
form during the process of carrying the baro- 
meter up the mountain; or, that the two 
observations at the first and last stations are 
made simultaneously by two observers. 

We have hitherto considered single in- 
tegration. Just as it is possible, and often 
either necessary or convenient, in the dif- 
ferential calculus to employ asecond differ- 
ential co-efficient, or the differential of a 
differential, so, in the inverse operation of 
the integral calculus, it is often either neces- 


188 THE INTERPRETATION OF 


sary or convenient to employ an integral of 
an intégral, or a double integral as it 1s called, 


and. which is indicated by the sign ||. 


As an example of the natural intro- 
duction of a double integral, let us consider 
a water-pipe carrying a stream of water, 
and suppose that it be required to deter- 
tnine the total quantity of water g, which 
flows through a pipe in a given time 4, 
from observations which determine the 
flow or velocity of movement of water, at 
different points of the. cross-section of the 
pipe. In other words, we are supposed to 
know the velocity at different points of 
the cross-section and not to know the 
average velocity or total flow. [If the 
velocity continued uniform at every point, 
it would only be necessary to find, from the 
velocity at each point, the average velocity, 
and this would give us at once the total 


MATHEMATICAL FORMULA. 189 


flow, because if A, be the area of cross- 
section of the pipe, in square centimetres, 
and v, the mean velocity in centimetres- 
per-ssecond; then the volume of water 
flowing in each second will be 


V = Av cubic centimetres, or grammes, 
of water per second. 


It is evident that a single integration 
will enable us to determine the average 
velocity v, from the assumed knowledge 
of the velocity at each point in the cross- | 
section. But if we suppose that the 
velocity is not uniform, but varies accord- 
ing to an assigned law, not only at differ- 
ent points of the cross-section but also at 
different times, then we shall have to 
determine not only the average velocity of 
the cross-section at any moment by the 
single integration, but also the average 
velocity in time by a second integration, 


190 THE INTERPRETATION OF 


and the solution of the problem may, 
therefore, be capable of expression as a 
double integration, in which one integra- 
tion refers to space and the other to time. 

When, therefore, such an equation is 
presented as the following 

F= fal A dw dy, 
Yid X1 

it means that the quantity /} is a double 
integral, or the integral of an integral. 

The equation may be represented as 
follows: 


ies es I A de | dy 
Agee Xi 


or, representing the quantity inside the 
bracket by BS, 


- F\” Bay. 
Y1 


If, therefore, we integrate the quantity A, 
with respect to @ alone, that is consider: 


MATHEMATICAL FORMULA. 191 


ine @, as an independent variable, and 
denote by 4, the integral so obtained, and 
then integrate the quantity 5, with re- 
spect to y, alone, considering y, the inde- 
pendent variable, the result of the second 
integration will give the quantity / 

In the same way triple, quadruple or 
higher integrals may be regarded as a 
succession of integrals, one being taken at 
atime. ‘The area of a plane figure whose 
sides conform to a definite geometrical 
law can be usually expressed as a double 
integral, the first integral being taken with 
respect to a, a length parallel to one axis 
of co-ordinates, and the second integral 
being taken with respect to y, a length 
parallel to the other axis of co-ordinates. 
Similarly, the volume of a figure bounded 
by outlines which are defined by any 
geometrical law can usually be expressed 
as a triple integral, the first integral being 


192 MATHEMATICAL FORMULA. 


taken with respect to a, the second with 
respect to y, and the third with respect 
to z In other words, the volume is 
expressed as equal to an indefinitely great 
number of little elements of volume, each 
of which has a cube of dimensions dx x 
dy x dz. Thus the equation 


V= \\| A dx dy dz 


represents the simplest form of a volume 
or triple integral. In practice, triple inte- 
gration is, usually, as far as multiple 


integration extends. 


CHAPTER XIII 
DETERMINANTS. 


Tue following pair of symmetrical equa- 
tions, 
Oa =D (A) 
URS OY nO, 


are called s¢multaneous equations involving 
two unknown quantities; namely, w and 
y. From these two equations both of the 
unknown quantities may be computed. 

Similarly, from the three following sim- 
ultaneous equations, 


4¢ + 2y + 82 = 17 
LO 22 = 3 (D) 
e+ Oy — 42 = %, 

193 


194 THE INTERPRETATION OF 


each of the three unknowns a@, y, and 2, 
can be computed; and, in general, any set 
of m, simultaneous equations, of the first 
degree, all of which are independent of 
each other; 2. ¢, are not reducible one to 
another, so that each contains an inde- 
pendent statement,—are sufficient for the 
determination of x unknown quantities. 

When only two simultaneous equations 
have to be dealt with, as in the pair given 
above, the computation is a simple matter, 
and, in some cases, it can be carried out 
by mere inspection. It is easy to see that 
the pair of above equations are satisfied by 
the results, 


Drea nt sane 
since 
8X1+2=5 
2 X13 XD ae 


When, however, a set of simultaneous 


MATHEMATICAL FORMULA. 195 


equations containing more than three un- 
knowns is given, the computation, while 
not necessarily difficult, is often very 
tedious and lengthy. It has been found 
that the process may be conducted in a 
regular way, which can be formulated by a 
sort of algebraic shorthand, which enables 
the process to be carefully inspected, 
checked, and often simplified. This proc- 
ess has led to the introduction and use of 
what are called determinants. 

A. determinant consists of a symmetrical 
assemblage of quantities in rows and col- 
ums, bounded by a pair of vertical lines, 
there being as many rows as there are 
columns. Thus: 


ag 


fst] |e | [os 


Beet 











are determinants of the second order, 


196 THE INTERPRETATION OF 


because each determinant has two rows 
and two columns. Similarly, 


4° 9" 2 h-o w 6) 4s 
Denny Talreeo ao ae 
uh pla: Sn henge 3° Om 


are determinants of the third order, be- 
cause each has three rows and three 
columns. Again, 


16 40058 2 a 
O 22) 20) Lae 
15° 46. 94) 23 aa, 
24.18 12 10 1 
GAD c2 15 ae 


is a determinant of the fifth order, because 
it has five rows and five columns. 
The separate numbers, or symbols, in a 


MATHEMATICAL FORMULZ. 197 
determinant ; or, as they are called, the ede- 
ments, are to be multiplied according to a 
definite rule. A determinant of the second 
order naturally produces terms or products 
having two elements each. A determinant 
of the third order naturally produces terms 
or products of three elements each ; and a 
determinant of the mth order naturally 


produces terms or products of 2 elements, 


each. Thus: 


abe 


aef 
ght 


= aet + dhe + gbf — gec —hfa — db, 








or the determinant is identical with the 
product : 


aev + dhe + gbf — gec — hfa — idb. 


This determinant of the third order is, 


therefore, a brief method of expressing 


198 THE INTERPRETATION OF 


these six terms, each containing products 
of three elements. ‘The terms are obtained 
by taking one element in each horizontal 
row, and combining with it one letter in 
each of all the other rows. Thus, the first 
term ae, takes the first element of the first 
row, the second element of the second row, 
and the third element of the third row, 
affixing to the same the positive sign. 
The fifth term has the first letter of the 
first row, the third letter of the second 
row, and the second letter of the third 
row, prefixing to the product the negative 
sign. In this way each term has one ele- 
ment, and only one element, out of each 
row, and takes the positive or negative 
sign according to the way in which the 
selection is made, following a definite rule. 

A. complete determinant; t. €, a deter- 
minant which has no zeros in it, of the 
2d order, is identically equivalent to two 


MATHEMATICAL FORMULAE. 199 


terms, each consisting of the product of 
two elements, 

A complete determinant of the third 
order is identically equivalent to six terms, 
each consisting of the product of three 
elements. 

A complete determinant of the fourth 
order is identically equivalent to twenty- 
four terms, each consisting of the product 
of four elements. 

A complete determinant of the fifth 
order is identically equivalent to 120 
terms, each consisting of the product of 
five elements. 

A. complete determinant of the nth 
order is equivalent to n! terms, each con- 
sisting of the product of m elements. 

The application of determinants may 
be illustrated sufficiently for our present 
purposes by considering the three simulta- 
neous equations (5). 


900 THE INTERPRETATION OF 


The value of the unknown quantity a, 
in these three equations, is obtained by 
the quotient of two determinants; namely, 





12 oe hes 
8 1-2 
pow 9 =4| 2 3 
As aeaee 294 
foi Wis -8 
1 9-4 


Similarly, the value of y, is the ratio of 
another pair of determinants: 


eT RR aT 
| 
cw 
Or 
oe) 
co 


MATHEMATICAL FORMULE. 201 


and, finally, the value of z, is the ratio of a 
third set-of determinants ; namely, 





Ae 17 
feria 

eo tel 59 SSE 
A598 945: 
eyelis 2 
1 9-4 


Each of the determinants in the nume- 
rator and denominator may be worked out 
by the equation already given for a deter- 
minant of the 3d order. The student will 
observe that the determinants forming the 
* denominators of the above fractions are all 
the same, and that the determinants form- 
ing the numerators differ from each other 
in a manner which is readily traced by 
reference to the coefficients in the three 
simultaneous equations (/). 


202 MATHEMATICAL FORMULA. 


There are various rules for treating, 
simplifying and reducing determinants, 
but it will be sufficient for the student to 
remember that a determinant is an abbrev1- 
ated form of a symmetrical set of products 
such as commonly presents itself in the 
process of solving simultaneous equations, 


CHAPTER XIV. 


SYNOPSIS OF SYMBOLS COMMONLY FOUND IN 
MATHEMATICAL FORMULA. 


qe Plus, or sign of addition. 5+ 7 =12. 

— Minus, or sign of subtraction. 7 — 
5 = 2. 

+ Plusorminus. 7+5=12 or 2. 

~ Difference sign. 5~%7=+2. The 
difference between 5 and 7 is 2. 

> Greater than. 7>5; seven is 


greater than five. 
< Less than. 5< 7; five is less than 


seven. 
= Hquality 5+7=12. 
+ Inequality. 5+ 7; five is not equal 


to seven: or, 7 and 5 are unequal. (2). 
203 


204 THE INTERPRETATION OF 


= Identity. 2(a@+ 6) =2a+ 26; both 
members invariably identically equal. 

2 Equality or Superiority. f 2 80; f 
greater than or equal to 30. 

S Equality or Inferiority. f $30; f 
less than or equal to 30. 

~ Nearly equal to. 0.667 ~ 4. 

x Multiplication. 5 x 7 = 35. 

. Multiplication. @.b=axX b= ab. 

+ Division. 3+ 4= 0.75. 


ae 3 
— Division bar. aes 83+4=0.75. 


/ Division solidus. 38/4=8+4=0.%5. 


() Brackets’ or Parentheses. 5(6+ 7) 


= 5{6+71=5[6+7]=5 X13, 


| Vinculum lines 5 X6+7+8 
=5(6+74+8)=5 X21 = 105, 
_o, Infinity. 8 xX 38X8....ad infin 


tum=o. 


MATHEMATICAL FORMUL. 205 


«, Varies as. Pressure of a liquid col- 
umn « the depth of liquid. 

.*., Lherefore. Because 3+5=8 .°. 
$+3+5=3+48. | 

me Mnces ot ot oo + 8.68 TH 
= 8. 

- 3: :, Proportionality. 3: 5::6 : 10; 
three is to five, as is six to ten. 

*, Square. 3° = 9, 

® Cube. 3? = 2%. 

", Index or Exponent. 38"=38 X 8 X 8, 
m times in all. 


-*, Negative Index. 3-"= 


*, Surd Exponent. 4? = square root 
of 4 = 2. 


m , Fractional Exponent. 4 = nth root 
of 4. 

Jor ¥, Radical or Surd. V4=42=2 
= square root of 4. 


veCube Root, V¥27 =3;3 V729 = 9, 


206 THE INTERPRETATION OF 


Vv, nth Root. Va= ai = nth root of a. 
!or L, Factorial T!=7 xX 6xX5x4xX 
LD ace enn 

=, Summation. (abe) = sum of all 


terms of the type ade. 











, Determinant. ; 7 = AX as eee 
a . Circumference 
7, Bi (Greek). Ratio ~ ‘Diamerenn = 
Blt LOU I gee 
log, 2, Common Logarithm of @. 


logy) 100 = 2.000. 

log. x, or hyp. log. v, Naperian Logarithm 
of a, log, 100 = 4.6052. 

Zor j, Sign of the imaginary. ¢@=7 = 
V—1, 

f(a) Funetion of « «*, V2, log a, am, 
are functions of « denotable by 7(@). 

sin a, Sine of angle a, Perpendicular + 


hypothenuse, 


MATHEMATICAL FORMULA. YO7T 


cos a, Cosine of angle a Base + hy- 
pothenuse. 

tan a, Tangent of angle a. Perpendicu- 
lar + base. 

cot a, Cotangent of angle a. Reciprocal 
of tangent. ; 

sec a, Secant of angle a Reciprocal of 
cosine. 

cosec a, Cosecant of angle a Recipro- 
eal of sine. 

vers a, Versed sine of anglea, 1 — cos a. 

sin-’ a, Inverse sine. The angle whose 
sine is alpha. 

sinh a, Hyperbolic sine of angle a. 

cosh a, Hyperbolic cosine of angle a. 

tanh a, Hyperbolic tangent of angle a. 
sinh a/cosh a. 

coth a, Hyperbolic cotangent of angle a. 
1/tanh a. 

sech a, Hyperbolic secant of angle a. 
1/cosh a. 


908 THE INTERPRETATION OF 


cosech a, Hyperbolic cosecant of angle a. 
1/sinh a. 
Ay, Difference of ¥. 
dy, Differential of y. Limit of sy. 
gi Differential coefficient of y, with re- 
wv 
hehe ts , AY 
spect to a Limiting ratio of rr when 4@ 
= 0, 


a ; : : 
mae Second differential coefficient of y, 


(a) 
with respect to 2. da) - 
di 


d° eerie RAL 
on) nth differential coefficient of y to @. 





y, Differential of y, with respect to 
dy 
time. dt : 

y, Differential of y, with respect to 


d 
space. - ; 


MATHEMATICAL FORMULA. 209 
| Integration sign. 
| J (@)da, Integral of f(@), with respect 


to @. 


|| Double Integral. 


| | i Triple Integral. 


F 
ae! 
w 


in 


i 





INDEX. 


Algebra, Definition of, 1. 
Algebraic and Arithmetical Equations, Difference 
between, 5-8. 
Equation, 5. 
Angle, Cosecant of, 107, 
, Cosine of, 104. 
, Cotangent of, 105. 
, First Quadrant of, 100. 
, Fourth Quadrant of, 101, 
, Hyperbolic, 129. 
, Positive, 100. 
, Secant of, 106. 
, second Quadrant of, 101. - 
, Sine of, 103. 
, Pangent of, 104. 
, Third Quadrant of, 101. 
Apparent Time at Sea, Trigonometrical Formula 
for, 119, 124. 
Application of Differential Calculus to the Laws 
of Falling Bodies, 141-148. 
211 






































912 INDEX. 


Application of Integral Calculus to the Laws of 
Falling Bodies, 170-176. 

Applications of Logarithms, 84-89. 

of Logarithms in Involution and Evyolu- 
tion, 90-93. . 

of Second Differential Coefficient, 159- 
164, . 

Arithmetical Equation, 4. 

: Series, Formula for, 32. 

Asymptote of Curve, 130, 











B 
Bar, Division, 35. 
Barometric Observations of Height, Application of 
Calculus to, 181. 
Base, Logarithmic, 78. 
Bracket, 27. 


C 


Calculus, Differential, 135. 

, Integral, 165. 
Cancellation, 74. 

Characteristic of Logarithm, 81. 
Circular Arc, Degrees of, 96. 
Arc, Minutes of, 96. 
Arc, Seconds of, 96. 
Functions, 102. 








' 








INDEX. 213 


Coefficient, Differential, 140. 

, Second Differential, 158. 
, Lhird Differential, 158. 
Common Logarithms, 79. 
Complete Determinant, 198. 
Compound Terms, 36. 
Conduction of Heat, Formula for, 39, 40. 
Cosecant of Angle, 107. 

Cosine, Hyperbolic, 131. 

Cosine of Angle, 104. 
Cotangent of Angle, 105, 
Counter-Clockwise Motion, 100. 
Cube of a Number, 50. 

Cube Root, 62. 

Curve, Asymptote of, 130. 








D 


Decimal Part of Logarithm, 81. 

Definition of Algebra, 1. 

of Equation, 2. 

Deflection of Horizontal Beam Supported at One 
End, Formula for, 188, 134. 

Degrees of Circular Are, 96. 

Dependent Variables, 150. 

Determinant, 198. 

, Elements of, 196. 

, Sign for, 206. 











914 INDEX. 


Determinants, 193. 

of the Second Order, 197. 

of the Third Order, 197. 

Difference between Algebraic and Arithmetical 
Equations, 5-8, 

Sign, 203. 

Differential, 140. 

Calculus, 135. 

Calculus, Nature of, 135-139. 

Coefficient, 140. 

Distance Passed through by Freely Falling Body, 

Formula for, 58. 

Division, 34. 

Bar, 35. 

, sign of, 34. 

Double Integral, Sign for, 209. 

Integrals, 188. 

— Integration, 188. 


E 


Elements of Determinant, 196. 
Equality Sign, 2. 
Equation, Algebraic, 5. 
, Arithmetical, 4. 
———.,, Definition of, 2. 

——, Left-Hand Side of, 8, 
, Number of Terms of, 9. 
, Quadratic, 71. 









































INDEX. 215 


Equation, Right-Hand Side of, 3. 

, Simple, 71. 

Equations, 70. 

of the Fourth Degree, 73, 74. 

of the Third Degree, 72. 

, Simultaneous, 193. 

, Lrigonometrical, 117, 118. 

Evolution, 61. 

and Involution, Application of Logarithms 
in, 90-93. 

Expansion of a Gas, Formula for, 37, 38. 

Exponent of Number of Quantity, 51. 




















F 

Factorial, 48. 

—, Sign for, 206. 

First Quadrant of Angle, 100. 

Formula for Arithimetical Series, 32. 

for Conduction of Heat, 39, 40. 

for Deflection of Horizontal Beam Sup- 
ported at One End, 133, 134. 

for Distance Passed through by Freely 
Falling Body, 58. 

for Expansion of a Gas, 37, 38. 

for Gravitational Force at Different Lati- 
tudes, 117, 118. 

for Horse-Power or Single-Cylinder 
Engine, 43, 























216 INDEX. 


Formula for Joint Resistance, 39. 

for Period of Time of Complete Vibration 
of Simple Pendulum, 63, 64. 

for Surface of Right Cylinder, 42. 

for Volume of Liquid at Given Tempera- 
ture, 75. 

for Work Done by Volume of Expanding 
Gas at Constant Temperature, 95. 

Relating to Theory of Probabilities, 44, 
45. 

, Trigonometrical, 115. 

Fourth Degree, Equations of, 73, 74. 

Power of a Number, 51. 

— Quadrant of Angle, 101. 

Root, 62. 

Fractional Indices, 66-69. 

Functions, Circular, 102. 

, Hyperbolic, 132. 

, Hyperbolic Trigonometrical, 126. 

, Trigonometrical, 102. 



































H 


Horse-Power of Single Cylinder Engine, Formula 
for, 43. 

Hyperbola, Rectangular, 128, 

Hyperbolic Angle, 129. 

Cosine, 131. 





INDEX. yA bras 


Hyperbolic Functions, 132. 
Logarithms, 94. 

Sine, 130. 

Trigonometrical Functions, 126. 











Identity, Sign of, 204. 

Independent Variables, 151. 

Index of a Number, 51. 

Indices, Fractional, 66-69. 

, Positive and Negative, 54-58. 

Inequality, Sign of, 202. 

Inferiority, Sign of, 204. 

‘Infinity, Sign of, 204. 

Initial Line of Radius Vector, 102. 

Instantaneous Velocity of Falling Body, 148. 

Integral Calculus, 165. 

Calculus, Application of, to the Laws of 
Falling Bodies, 170-176. 

Calculus, Scope of, 166-168. 

Integrals, Double 188. 

, Friple, 192. 

Integrating Meter, 177-179. 

Integration, Double, 188. 

Sign, 209. 

Inverse Notation, 126. 

Inyolution, 50, 























218 INDEX. 


Involution and Evolution, Application of Loga- 
rithms in, 90-93. 


J 


Joint Resistance, 39. 


L 


Law of Summation of Indices, 54. 

Laws of Falling Bodies, Application of Differen- 
tial Calculus to, 141-148. 

Left-Hand Member of Equation, 3. 

Logarithm, Characteristic of, 81. 

, Decimal Part of, 81. 

, Mantissa of, 81. 

, Negative Characteristic of, 125. 

Logarithmic Base, 78. 

Logarithms, 78. 

, Applications of, 84-89. 

, Common, 79. 

, Hyperbolic, 94. 

———., Naperian, 94. 

, Natural, 94. 

, Tables of, 80. 


























M 


Mantissa of Logarithm, 81. 
Meter, Integrating, 177-179, 


INDEX. 219 


Mathematical Formula for Number of Balls in 
Square Pyramid of Cannon Ball, 46. 

Formule, Symbols Commonly Found in, 
203-209. 

Minus Sign, 14. 

Minutes of Circular Arce, 96. 

Motion, Counter-Clockwise, 100. 

, Negative, 100. 

, Positive, 100. 

Multiplication, 21. 

Sign, 21. 

Sign, Omission of, 24. 

——— Sign, Use of Period for, 25. 

















N 


Naperian Logarithms, 94. 

Natural Logarithms, 94. 

Nature of Differential Calculus, 185-139. 
Negative Characteristic of Logarithm, 125. 
Index, Sign for, 205. 

Motion, 100. 

Remainder, 15. 

7 Terms, 16-18. 

Notation, Inverse, 126. 

Nth Power of a Number, 51, 

Nth root, 62. 

Number, Cube of, 50. 

















220 | INDEX. 


Number, Fourth Power of, 51. 

, Index of, 51. 

——., Nth Power of, 51. 

of Terms of Equation, 9. 

or Quantity, Exponent of, 51. 
, Square of, 50. 

, Powers of, 50. 

















O 


Omission of Multiplication Sign, 24. 


li 


Parenthesis, 27. | 

Plus Sign, 2. 

Positive and Negative Indices, Use of, 54-58. 
Angle, 100. 

Motion, 100. 

Power, Second, 50. 

, Third, 50. 

Powers of Numbers, 50. 

Product, 21. 











Q 


Quadratic Equation, 71. 
Quantity of Magnetic Flux, Application of Calcu- 
lus to the Determination of, 148, 150. 


INDEX. 291 


Quantity of Water Discharged from a Reservoir, 
Application of Calculus to the Determi- 
nation of, 145, 147. 

Quotient, 34. 


R 


Radian, 96. 

Radical Sign, 63. 

Radius Vector, 99. 

Vector, Initial Line of, 102. 
Rectangular Hyperbola, 128. 
Remainder, Negative, 15. 
Right-Hand Member of Equation, 3. 
Root, Cube, 62. 

, Fourth, 62. 

——., Nth, 62. 

, Square, 61. 

, Third, 62. 

Roots, 61. 














S 


Scope of Integral Calculus, 166-168. 

Secant of Angle, 106. 

Second Differential Coefficient, 158. 

Differential Coefficient, Applications of, 
159-164. 

Power, 50. 

Quadrant of Angle, 101. 

Seconds of Circular Arc, 96. 











959 INDEX. 


Simple Equation, 71. 

Pendulum, Formula for Period of Time of 
Complete Vibration of, 63, 64. 

Sign, Difference, 203. 

, Equality, 2. 

for Determinant, 206. 
for Double Integral, 209. 
for Factorial, 206. 

for Negative Index, 205. 
for “ Since,” 205. 

for Summation, 206. 

for Surd, 205. 

for “ Therefore,” 205. 
for Triple Integral, 209. 
for “ Varies as,” 204. 

-—_——.,, Minus, 14. 

, Multiplication, 21. 

of Division, 34. 

ef Identity, 204. 

of Inequality, 202. 

of Inferiority, 204. 

of Infinity, 204. 

of Integration, 209. 

of Superiority, 204. 

of the Imaginary, 206. 

» Plus, 2. 

, Radical, 63. 

Simultaneous Equations, 193. 

“ Since,” Sign for, 205. 







































































INDEX. Bee 


Sine, Hyperbolic, 130. 

-of Angle, 1038. 

Solidus, 35. 

Square of a Number, 50 

Pyramid of Cannon Balls, Mathematical 
Formula for Number of Balls in, 46, 

Root, 61. 

Subtraction, 14. 

Sum of All the Terms, Symbol for, 47. 

Summation of Indices, Law of, 54. 

, Sign for, 206. 

Superiority, Sign of, 204. 

Surd, Sign for, 205. 

Surface of Right Cylinder, Formula for, 42. 

Symbol for Sum of All the Terms, 47. 

Symbols, 10 

Commonly Found in Mathematical For- 
mul, 203-209. 


T 


Tables of Logarithms. 80. 

Tangent of Angle, 104. 

Terms, Compound, 36. 

, Negative, 16-18. 

of Equation, 9. 

“S'herefore,” Sign for, 205. 

Theory of Probabilities, Formule Relating to, 44, 
4, 























os): INDEX. 


Thermal Capacity of a Body, Applications of 
Calculus to Determination of, 147-149. 

Third Degree, Equations of, 73. 

Differential Coefficient, 158. 

Power, 50. 

Quadrant of Angle, 101. 

Root, 62. 

Trigonometrical Equations, 117, 118. 

Formula for Apparent Time at Sea, 119, 
124, 

Formule, 115. 

Functions, 102. 

Functions, Variations in Value of, with 
Variations of Angle, 108-114. 

Trigonometry, 97. 

Triple Integral, Sign for, 209. 

Integrals, 192. 





























U 


Use of Period for Multiplication Sign, 25. 
of Vinculum, 30. 





Vv 


Variables, Dependent, 150. 
——, Independent, 151. | 
Variations of Trigonometrical Functions with 
Variations of Angle, 108-114. 


% 





INDEX, 225 


Vector, Radius, 99. 

Velocity, Instantaneous, of Falling Body, 143. 

Vinculum, Use of, 30. 

Volume of Liquid at Given Temperature, Formula 
Fors. 75; 


W 


Whole Number or Characteristic of Logarithm, 81. 
Work Done by Volume of Expanding Gas at 
Constant Temperature, Formula for, 95. 


THE END. 











pres 


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